Topics in Absolute Anabelian Geometry I

8/13/2019 Topics in Absolute Anabelian Geometry I

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TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I:

GENERALITIES

Shinichi Mochizuki

March 2012

Abstract. This paper forms the first part of a three-part series in which we treatvarious topics in absolute anabelian geometry from the point of view of develop-ing abstract algorithms, or software, that may be applied to abstract profinitegroups that just happen to arise as [quotients of ] etale fundamental groups fromalgebraic geometry. One central theme of the present paper is the issue of under-standing the gap between relative, semi-absolute, and absolute anabeliangeometry. We begin by studying various abstract combinatorial propertiesof profi-nite groups that typically arise as absolute Galois groups or arithmetic/geometricfundamental groups in the anabelian geometry of quite general varieties in arbitrarydimension over number fields, mixed-characteristic local fields, or finite fields. Theseconsiderations, combined with the classical theory of Albanese varieties, allow usto derive an absolute anabelian algorithmfor constructing the quotient of anarithmetic fundamental group determined by the absolute Galois group of thebase field in the case ofquite general varieties of arbitrary dimension. Next, wetake a more detailed look at certain p-adic Hodge-theoretic aspects of the absolute

Galois groups of mixed-characteristic local fields. This allows us, for instance, toderive, from a certain result communicated orally to the author by A. Tamagawa, asemi-absolute Hom-version whose absolute analogue is false! of the an-abelian conjecture for hyperbolic curves over mixed-characteristic local fields.Finally, we generalize to the case ofvarieties ofarbitrary dimension over arbitrarysub-p-adic fields certain techniques developed by the author in previous papers overmixed-characteristic local fields for applying relative anabelian results to obtainsemi-absolute group-theoretic contructionsof the etale fundamental groupof one hyperbolic curve from the etale fundamental group of another closely relatedhyperbolic curve.

Contents:

0. Notations and Conventions1. Some Profinite Group Theory2. Semi-absolute Anabelian Geometry3. Absolute Open Homomorphisms of Local Galois Groups4. Chains of Elementary OperationsAppendix: The Theory of Albanese Varieties

2000 Mathematical Subject Classification. Primary 14H30; Secondary 14H25.

Typeset by AMS-TEX

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2 SHINICHI MOCHIZUKI

Introduction

The present paper is the first in a series of three papers, in which we con-tinue our study ofabsolute anabelian geometryin the style of the following papers:[Mzk6], [Mzk7], [Mzk8], [Mzk9], [Mzk10], [Mzk11], [Mzk13]. IfXis a [geometrically

integral]varietyover a field k, and Xdef= 1(X) is the etale fundamental group of

X[for some choice of basepoint], then roughly speaking,anabelian geometrymay be summarized as the study of the extent to which properties ofX suchas, for instance, the isomorphism classofX may be recovered from [variousquotients of] the profinite group X . One form of anabelian geometry is relativeanabelian geometry [cf., e.g., [Mzk3]], in which instead of starting from [vari-ous quotients of] the profinite group X , one starts from the profinite group Xequipped with the natural augmentation

X G

kto the absolute Galois group of

k. By contrast,absolute anabelian geometry refers to the study of proper-ties ofXas reflected solely in the profinite group X . Moreover, one may considervarious intermediate variants between relative and absolute anabelian geometrysuch as, for instance, semi-absolute anabelian geometry, which refers to thesituation in which one starts from the profinite group X equipped with the kernelof the natural augmentation X Gk.

Thenew point of viewthat underlies the various topics in absolute anabeliangeometry treated in the present three-part series may be summarized as follows.In the past, research in anabelian geometry typically centered around the establish-

ment offully faithfulness results i.e., Grothendieck Conjecture-type results concerning some sort of fundamental group functor X X from varietiesto profinite groups. In particular, the term group-theoretic was typically usedto refer to properties preserved, for instance, by some isomorphism of profinitegroups X

Y [i.e., between the etale fundamental groups of varieties X, Y].

By contrast:

In the present series, the focus of our attention is on the development ofalgorithms i.e., software which are group-theoretic in thesense that they are phrased inlanguagethat only depends on the structure

of the input dataas [for instance] a profinite group.

Here, the input data is a profinite group that just happens to arise from schemetheory as an etale fundamental group, but which is only of concern to us in itscapacity as an abstract profinite group. That is to say,

the algorithms in question allow one to construct various objects reminis-cent of objects that arise in scheme theory, but the issue of eventuallyreturning to scheme theory e.g., of showing that some isomor-phism of profinite groups arises from an isomorphism of schemes is no

longer an issue of primary interest.

One aspect of this new point of view is that the main results obtained are no longernecessarily of the form


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TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I 3

() some scheme is anabelian i.e., some sort of fundamental groupfunctor X X from varieties to profinite groups is fully faithful

but rather of the form

() some a priorischeme-theoretic property/construction/operationmay be formulated as a group-theoretic algorithm, i.e., an algorithmthat depends only on the topological group structureof the arithmetic fun-damental groups involved

cf., e.g., (2), (4) below. A sort of intermediate variant between () and () isconstituted by results of the form

() homomorphisms between arithmetic fundamental groups that satisfy

some sort ofrelatively mild conditionarise from scheme theory

cf., e.g., (3) below.

Here, we note that typically results in absolute or semi-absolute anabeliangeometry are much moredifficult to obtain than corresponding results in relativeanabelian geometry [cf., e.g., the discussion of (i) below]. This is one reason whyone is frequently obliged to content oneself with results of the form () or (), asopposed to ().

On the other hand, another aspect of this new point of view is that, by abol-

ishing the restriction that one must have as ones ultimate goal the complete recon-struction of the original schemes involved, one gains agreater degree of freedomin the geometries that one considers. This greater degree of freedom often results inthe discovery ofnew resultsthat might have eluded ones attentionif one restrictsoneself to obtaining results of the form (). Indeed, this phenomenon may alreadybe seen in previous work of the author:

(i) In [Mzk6], Proposition 1.2.1 [and its proof], various group-theoretic al-gorithms are given for constructing various objects associated to the ab-solute Galois group of a mixed-characteristic local field. In this case, werecall that it is well-known [cf., e.g., [NSW], the Closing Remark precedingTheorem 12.2.7] that in general, there exist isomorphisms between suchabsolute Galois groups that do not arise from scheme theory.

(ii) In the theory of pro-l cuspidalizations given in [Mzk13], 3, cuspi-dalized geometrically pro-l fundamental groups are group-theoreticallyconstructed from geometrically pro-l fundamental groups of proper hy-perbolic curves without ever addressing the issue of whether or not theoriginal curve [i.e., scheme] may be reconstructed from the given geomet-rically pro-l fundamental group [of a proper hyperbolic curve].

(iii) In some sense, the abstract, algorithmic point of view discussed aboveis taken even further in [Mzk12], where one works with certain types ofpurely combinatorial objects i.e., semi-graphs of anabelioids whosedefinition just happens to be motivated by stable curves in algebraic ge-ometry. On the other hand, the results obtained in [Mzk12] are results


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concerning theabstract combinatorial geometryof these abstract combina-torial objects i.e., one is never concerned with the issue of eventuallyreturning to, for instance, scheme-theoretic morphisms.

Themain resultsof the present paper are, to a substantial extent, generalitiesthat will be of use to us in the further development of the theory in the lattertwo papers of the present three-part series. These main results center around thetheme of understanding the gap betweenrelative, semi-absolute, and absoluteanabelian geometryand may be summarized as follows:

(1) In 1, we study various notions associated to abstract profinite groupssuch asRTF-quotients[i.e., quotients obtained by successive formationof torsion-free abelianizations cf. Definition 1.1, (i)], slimness [i.e.,the property that all open subgroups are center-free], and elasticity [i.e.,the property that every nontrivial topologically finitely generated closednormal subgroup of an open subgroup is itself open cf. Definition 1.1,(ii)] in the context of the absolute Galois groupsthat typically appearin anabelian geometry [cf. Proposition 1.5, Theorem 1.7].

(2) In2, we begin by formulating the terminologythat we shall use in ourdiscussion of the anabelian geometry of quite general varieties of arbitrarydimension [cf. Definition 2.1]. We then apply the theory of slimness andelasticity developed in1 to study various variants of the notion ofsemi-absoluteness[cf. Proposition 2.5]. Moreover, in the case ofquite generalvarieties of arbitrary dimensionover number fields, mixed-characteristiclocal fields, or finite fields, we combine the various group-theoretic consid-erations of (1) with the classical theory of Albanese varieties [reviewed inthe Appendix] to give various

group-theoretic algorithmsfor constructing the quotient of anarithmetic fundamental group determined by the absoluteGalois groupof the base field [cf. Theorem 2.6, Corollary 2.8].

Finally, in the case ofhyperbolic orbicurves, we apply the theory ofmax-imal pro-RTF-quotients developed in 1 to give quite explicit group-

theoretic algorithms for constructing these quotients [cf. Theorem 2.11].Such maximal pro-RTF-quotients may be thought of as a sort of analogue,in the case of mixed-characteristic local fields, of the reconstruction, in thecase of finite fields, of the quotient of an arithmetic fundamental groupdetermined by the absolute Galois group of the base field via the opera-tion ofpassing to the maximal torsion-free abelian quotient [cf. Remark2.11.1].

(3) In 3, we develop a generalization of the main result of [Mzk1] concerning

the geometricity of arbitrary isomorphisms of absolute Ga-

lois groups of mixed-characteristic local fields that pre-serve the ramification filtration [cf. Theorem 3.5].

This generalization allows one to replace the condition of preservingthe ramification filtration by various more general conditions, certain


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of which were motivated by a result orally communicated to the authorby A. Tamagawa [cf. Remark 3.8.1]. Moreover, unlike the main result of[Mzk1], this generalization may be applied, in certain cases, to

arbitrary open homomorphisms i.e., not just isomor-phisms!

between absolute Galois groups of mixed-characteristic local fields, henceimplies certain semi-absoluteHom-versions [cf. Corollary 3.8, 3.9] of therelative Hom-versions of the Grothendieck Conjecture given in [Mzk3],Theorems A, B. Also, we observe, in Example 2.13, that the correspondingabsolute Hom-versionof these results is false in general. Indeed, it wasprecisely the discovery of this counterexample to the absolute Hom-versionthat led the author to the detailed investigation of the gap be-

tween absolute and semi-absolute that forms the content of2.

(4) In 4, we study various fundamental operations for passing from onealgebraic stack to another. In the case of arbitrary dimension, these op-erations are the operations of passing to a finite etale coveringand passing to a finite etale quotient; in the case of hyperbolicorbicurves, we also consider the operations offorgetting a cusp andcoarsifying a non-scheme-like point. Our main result asserts that

if one assumes certain relative anabelian results concerningthe varieties under consideration, then there exist group-theoreticalgorithmsfor describing the corresponding semi-absolute an-abelian operations on arithmetic fundamental groups [cf. The-orem 4.7].

This theory, which generalizes the theory of [Mzk9], 2, and [Mzk13], 2,may be applied not only to hyperbolic orbicurves over sub-p-adic fields[cf.Example 4.8], but also to iso-poly-hyperbolic orbisurfaces over sub-p-adic fields[cf. Example 4.9]. In [Mzk15], this theory will be applied, in anessential way, in our development of the theory ofBelyiandelliptic cusp-idalizations. We also give a tempered versionof this theory [cf. Theorem

4.12].

Finally, in an Appendix, we review, for lack of an appropriate reference, various well-known facts concerning the theory ofAlbanese varietiesthat will play an importantrole in the portion of the theory of2 concerning varieties of arbitrary dimension.Much of this theory of Albanese varieties is contained in such classical referencesas [NS], [Serre1], [Chev], which are written from a somewhat classical point ofview. Thus, in the Appendix, we give a modern scheme-theoretic treatmentof thisclassical theory, but without resorting to the introduction ofmotives and derivedcategories, as in [BS], [SS]. In fact, strictly speaking, in the proofs that appearin the body of the text [i.e., 2], we shall only make essential use of the portionof the Appendix concerning abelian Albanese varieties [i.e., as opposed to semi-abelian Albanese varieties]. Nevertheless, we decided to give a full treatment ofthe theory of Albanese varieties as given in the Appendix, since it seemed to theauthor that the theory is not much more difficult and, moreover, assumes a much


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more natural form when formulated for open[i.e., not necessarily proper] varieties[which, roughly speaking, correspond to the case of semi-abelian Albanese varieties]than when formulated only forpropervarieties [which, roughly speaking, correspond

to the case of abelian Albanese varieties].

Acknowledgements:

I would like to thank Akio Tamagawafor many helpful discussions concerningthe material presented in this paper. Also, I would like to thank Brian Conradforinforming me of the references in the Appendix to [FGA], and Noboru Nakayamafor advice concerning non-smooth normal algebraic varieties.


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Section 0: Notations and Conventions

Numbers:

The notation Q will be used to denote the field of rational numbers. Thenotation Z Q will be used to denote the set, group, or ring ofrational integers.The notation N Z will be used to denote the set or monoid ofnonnegative rationalintegers. The profinite completionof the group Zwill be denotedZ. Write

Primes

for the set of prime numbers. Ifp Primes, then the notation Qp (respectively,Zp) will be used to denote the p-adic completionofQ (respectively, Z). Also, we

shall writeZ()p Z

p

for the subgroup 1 +pZp Zp ifp > 2, 1 +p

2Zp Zp ifp = 2. Thus, we have

isomorphisms of topological groups

Z()p (Zp/Z

()p )

Zp; Z

()p

Zp

where the second isomorphism is the isomorphism determined by dividing the

p-adic logarithm by p if p > 2, or by p2 if p = 2; Zp/Z()p

Fp if p > 2,

Zp/Z()p

Z/pZ ifp= 2.

A finite field extension ofQ will be referred to as a number field, or NF, forshort. A finite field extension ofQp for some p Primes will be referred to as amixed-characteristic nonarchimedean local field, orMLF, for short. A field of finitecardinality will be referred to as a finite field, orFF, for short. We shall regard theset of symbols {NF, MLF, FF}as being equipped with a linear ordering

NF> MLF > FF

and refer to an element of this set of symbols as a field type.

Topological Groups:

Let G be a Hausdorff topological group, and HG a closed subgroup. Let uswrite

ZG(H)def= {g G | g h= h g, h H}

for the centralizerofH in G;

NG(H)def= {g G | g H g1 =H}

for the normalizerofH in G; and

CG(H)def= {g G | (g H g1)Hhas finite index in H, g H g1}

for the commensuratorofH in G. Note that: (i) ZG(H), NG(H) and CG(H) aresubgroups ofG; (ii) we have inclusionsH, ZG(H) NG(H) CG(H); (iii) H is


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normal in NG(H). IfH = NG(H) (respectively, H = CG(H)), then we shall saythat H is normally terminal (respectively, commensurably terminal) in G. Notethat ZG(H), NG(H) are always closed inG, while CG(H) is not necessarily closed

inG. Also, we shall write Z(G)def

= ZG(G) for the centerofG.

LetG be a topological group. Then [cf. [Mzk14],0] we shall refer to a normalopen subgroup HG such that the quotient group G/His a free discrete groupas co-free. We shall refer to a co-free subgroupHG as minimalif every co-freesubgroup ofG containsH. Thus, any minimal co-free subgroup ofG is necessarilyuniqueand characteristic.

We shall refer to a continuous homomorphism between topological groups asdense (respectively, of DOF-type [cf. [Mzk10], Definition 6.2, (iii)]; of OF-type) ifits image is dense (respectively, dense in some open subgroup of finite index; an

open subgroup of finite index). Let be a topological group; a normal closedsubgroupsuch that every characteristic open subgroup of finite index H admitsa minimal co-free subgroup Hco-fr H. Write for the profinite completionof .Let Qbe a quotient of profinite groups. Then we shall refer to as the (Q, )-co-free

completion of, or co-free completion of with respect to [the quotient ]Qand [the subgroup] where we shall often omit mention of when it isfixed throughout the discussion the inverse limit

Q/co-fr def

= limH ImQ(/Hco-fr

)

whereH ranges over the characteristic open subgroups of of finite index;Hco-fr is the closure of the image ofHco-fr in;Hco-frQ Q is the image ofHco-fr in Q; ImQ() denotes the image in Q/Hco-frQ of the group in parentheses.Thus, we have a natural dense homomorphism Q/co-fr.

We shall say that a profinite groupG is slimif for every open subgroupH G,the centralizer ZG(H) is trivial. Note that every finite normal closed subgroupN Gof a slim profinite group G is trivial. [Indeed, this follows by observing thatfor any normal open subgroupHG such thatNH= {1}, consideration of theinclusion N G/Hreveals that the conjugation action ofH on N is trivial, i.e.,that NZG(H) ={1}.]

We shall say that a profinite group G is decomposable if there exists an iso-morphism of profinite groupsH1 H2

G, whereH1, H2 are nontrivial profinite

groups. If a profinite group G is not decomposable, then we shall say that it isindecomposable.

We shall write Gab for the abelianizationof a profinite group G, i.e., the quo-tient ofG by the closure of the commutator subgroup ofG, and

Gab-t

for the torsion-free abelianizationofG, i.e., the quotient ofGab by the closure ofthe torsion subgroup ofGab. Note that the formation ofGab, Gab-t is functorialwith respect to arbitrary continuous homomorphisms of profinite groups.


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We shall denote the group of automorphisms of a profinite groupG by Aut(G).Conjugation by elements of G determines a homomorphism G Aut(G) whoseimage consists of the inner automorphisms of G. We shall denote by Out(G)

the quotient of Aut(G) by the [normal] subgroup consisting of the inner auto-morphisms. In particular, if G is center-free, then we have an exact sequence1 G Aut(G) Out(G) 1. If, moreover,Gistopologically finitely generated,then it follows immediately that the topology ofG admits a basis ofcharacteristicopen subgroups, which thus determine a topologyon Aut(G), Out(G) with respectto which the exact sequence 1 G Aut(G) Out(G) 1 may be regarded asan exact sequence ofprofinite groups.

Algebraic Stacks and Log Schemes:

We refer to [FC], Chapter I, 4.10, for a discussion of thecoarse spaceassociatedto an algebraic stack. We shall say that an algebraic stack is scheme-likeif it is, infact, a scheme. We shall say that an algebraic stack is generically scheme-like if itadmits an open dense substack which is a scheme.

We refer to [Kato] and the references given in [Kato] for basic facts and def-initions concerning log schemes. Here, we recall that the interiorof a log schemerefers to the largest open subscheme over which the log structure is trivial.

Curves:

We shall use the following terms, as they are defined in [Mzk13],0: hyperboliccurve, family of hyperbolic curves, cusp, tripod. Also, we refer to [Mzk6], the proofof Lemma 2.1; [Mzk6], the discussion following Lemma 2.1, for an explanation ofthe termsstable reduction and stable modelapplied to a hyperbolic curve overan MLF.

IfX is a generically scheme-like algebraic stackover a field k that admits afinite etale Galois covering Y X, where Y is a hyperbolic curve over a finiteextension ofk, then we shall refer to X as a hyperbolic orbicurve over k. [Thus,when k is of characteristic zero, this definition coincides with the definition of ahyperbolic orbicurve in [Mzk13], 0, and differs from, but is equivalent to, the

definition of a hyperbolic orbicurve given in [Mzk7], Definition 2.2, (ii). We referto [Mzk13], 0, for more on this equivalence.] Note that the notion of a cusp ofa hyperbolic curve given in [Mzk13], 0, generalizes immediately to the notion ofcusp of a hyperbolic orbicurve. IfX Y is a dominant morphism of hyperbolicorbicurves, then we shall refer toX Y as apartial coarsification morphismif themorphism induced by X Y on associated coarse spacesis an isomorphism.

Let X be a hyperbolic orbicurve over an algebraically closed field; denote itsetale fundamental group by X . We shall refer to the order of the [manifestly finite!]decomposition group in X of a closed point x ofX as the order ofx. We shallrefer to the [manifestly finite!] least common multiple of the orders of the closed

points ofX as the order ofX. Thus, it follows immediately from the definitionsthat X is a hyperbolic curveif and only if the order ofX is equal to 1.


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Section 1: Some Profinite Group Theory

We begin by discussing certain aspects of abstract profinite groups, as they

relate to the Galois groups of finite fields, mixed-characteristic nonarchimedeanlocal fields, and number fields. In the following, let G be a profinite group.

Definition 1.1.

(i) In the following, RTF is to be understood as an abbreviation for recur-sively torsion-free. IfHG is a normal open subgroup that arises as the kernelof a continuous surjection G Q, where Q is a finite abelian group, that factorsthrough the torsion-free abelianization G Gab-t ofG [cf. 0], then we shall referto (G, H) as an RTF-pair. If for some integer n 1, a sequence of open subgroups

Gn Gn1 . . . G1 G0 =G

ofG satisfies the condition that, for each nonnegative integer j n 1, (Gj , Gj+1)is an RTF-pair, then we shall refer to this sequence of open subgroups as an RTF-chain [from Gn to G]. If H G is an open subgroup such that there exists anRTF-chain fromH to G, then we shall refer to HG as an RTF-subgroup [ofG].If the kernel of a continuous surjection : G Q, where Q is a finite group, is anRTF-subgroup ofG, then we shall say that : G Q is an RTF-quotientofG.If: G Q is a continuous surjection of profinite groups such that the topology

of Q admits a basis of normal open subgroups {N}A satisfying the propertythat each composite G Q Q/N [for A] is an RTF-quotient, then weshall say that : G Q is a pro-RTF-quotient. If G is a profinite group suchthat the identity map ofG forms a pro-RTF-quotient, then we shall say that G isa pro-RTF-group. [Thus, every pro-RTF-group is pro-solvable.]

(ii) We shall say that G is elastic if it holds that every topologically finitelygenerated closed normal subgroup N H of an open subgroup H G of G iseither trivial or of finite index in G. IfG is elastic, but nottopologically finitelygenerated, then we shall say that G isvery elastic.

(iii) Let Primes [cf. 0] be a set of prime numbers. If G admits anopen subgroup which is pro-, then we shall say that G is almost pro-. Weshall refer to a quotient G Q as almost pro--maximalif for some normal opensubgroup N G with maximal pro- quotient N P, we have Ker(G Q) =Ker(N P). [Thus, any almost pro--maximal quotient ofG is almost pro-.]

If def

= Primes\ {p} for some p Primes, then we shall write pro-(=p) forpro-. Write Z(=p)for the maximal pro-(=p) quotientofZ. We shall say that G is pro-omissive(re-spectively, almost pro-omissive) if it is pro-(=p) for somep Primes(respectively,

if it admits a pro-omissive open subgroup). We shall say that G is augmented pro-pif there exists an exact sequence of profinite groups 1 N G Z(=p) 1,whereN ispro-p; in this case, the image ofN inGisuniquely determined[i.e., as the

maximal pro-psubgroup ofG]; the quotientG Z(=p) [which is well-defined up to


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automorphisms ofZ(=p)] will be referred to as the augmentationof the augmentedpro-pgroupG. We shall say that G isaugmented pro-primeif it is augmented pro-p for some [not necessarily unique!] p Primes. If = {p} for some unspecified

p Primes, we shall write pro-prime for pro-. IfC is the full formation[cf., e.g., [FJ], p. 343] of finite solvable -groups, then we shall refer to a pro-Cgroup as a pro--solvable group.

Proposition 1.2. (Basic Properties of Pro-RTF-quotients)Let

: G1 G2

be a continuous homomorphism of profinite groups. Then:

(i) IfH G2 is an RTF-subgroup ofG2, then1(H) G1 is an RTF-

subgroup ofG1.

(ii) IfH, JG are RTF-subgroups ofG, then so isH

J.

(iii) IfHG is an RTF-subgroupofG, then there exists anormal [open]RTF-subgroup JG ofG such thatJH.

(iv) Every RTF-quotientG Q ofG factors through the quotient

G GRTF def

= limN

G/N

where N ranges over the normal [open] RTF-subgroups of G. We shall referto this quotient G GRTF as the maximal pro-RTF-quotient. Finally, theprofinite group GRTF is a pro-RTF-group.

(v) There exists a commutative diagram

G1

G2

GRTF1

RTF

GRTF2

where the vertical arrows are the natural morphisms, and the continuous homo-morphismRTF is uniquely determined by the condition that the diagram commute.

Proof. Assertion (i) follows immediately from the definitions, together with thefunctorialityof the torsion-free abelianization [cf. 0]. To verify assertion (ii), oneobserves that an RTF-chain from H

J to G may be obtained by concatenating

an RTF-chain from H

J to J[whose existence follows from assertion (i) appliedto the natural inclusion homomorphism J G] with an RTF-chain from J toG. Assertion (iii) follows by applying assertion (ii) to some finite intersection ofconjugates ofH. Assertion (iv) follows immediately from assertions (ii), (iii), andthe definitions involved. Assertion (v) follows immediately from assertions (i), (iv).


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Proposition 1.3. (Basic Properties of Elasticity)

(i) LetH G be an open subgroup of the profinite group G. Then theelas-ticityofG implies that ofH. IfG isslim, then the elasticity ofHimplies that ofG.

(ii) Suppose that G isnontrivial. Then G isvery elastic if and only if itholds that every topologically finitely generated closed normal subgroup N H ofan open subgroup HG ofG is trivial.

Proof. Assertion (i) follows immediately from the definitions, together with thefact that a slim profinite group has no normal closed finite subgroups [cf. 0]. Thenecessityportion of assertion (ii) follows from the fact that the existence of a topo-logically finitely generated open subgroup ofG implies thatG itself is topologically

finitely generated; the sufficiency portion of assertion (ii) follows immediately by

taking N def= G = {1}.

Next, we consider Galois groups.

Definition 1.4. We shall refer to a fieldk as solvably closed if, for every finiteabelian field extension k ofk, it holds that k =k.

Remark 1.4.1. Note that ifk is a solvably closed Galois extensionof a field k oftype MLF or FF [cf. 0], thenk is an algebraic closureofk. Indeed, this followsfrom the well-known fact that the absolute Galois group of a field of type MLF orFF is pro-solvable[cf., e.g., [NSW], Chapter VII, 5].

Proposition 1.5. (Pro-RTF-quotients of MLF Galois Groups)Letk be an

algebraic closure of anMLF[cf. 0]k of residue characteristicp; Gkdef= Gal(k/k);

Gk GRTFk themaximal pro-RTF-quotient [cf. Proposition 1.2, (iv)] ofGk.

Then:

(i) GRTF

k isslim.

(ii) There exists an exact sequence 1 P GRTFk Z 1, whereP is a

pro-pgroupwhose image inGRTFk is equal to the image of the inertia subgroupofGk inG

RTFk . In particular,G

RTFk is augmented pro-p.

Proof. Recall fromlocal class field theory[cf., e.g., [Serre2]] that for any open sub-groupH Gk, corresponding to a subfield kHk, we have a natural isomorphism

(kH) Hab

[where the denotes the profinite completion of an abelian group; denotesthe group of units of a ring]; moreover, Hab fits into an exact sequence

1 OkH Hab Z 1


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[where OkH kH is the ring of integers] in which the image ofOkH

inHab coincideswith the image of the inertia subgroup ofH. Observe, moreover, that the quotientof the abelian profinite group OkH by its torsion subgroup is a pro-p group. Thus,

assertion (ii) follows immediately from this observation, together with the definitionof themaximal pro-RTF-quotient. Next, let us observe that by applying thenaturalisomorphism(OkH ) Qp

kH, it follows that whenever H is normal in Gk, the

action ofGk/H on Hab-t is faithful. Thus, assertion (i) follows immediately.

The following result is well-known.

Proposition 1.6. (Maximal Pro-pQuotients of MLF Galois Groups) Let

k be an algebraic closure of anMLFk of residue characteristicp; Gkdef= Gal(k/k);

Gk G(p)k themaximal pro-p-quotientofGk. Then:

(i) Anyalmost pro-p-maximal quotient Gk Q ofGk isslim.

(ii) Suppose further thatk contains aprimitive p-th root of unity. Then

for any finite moduleMannihilated byp equipped with a continuous action byG(p)k

[which thus determines a continuous action by Gk], the natural homomorphism

Gk G(p)k induces anisomorphism

Hj(G(p)k , M)

Hj(Gk, M)

on Galois cohomology modules for all integersj 0.

(iii) If k contains (respectively, does not contain) a primitive p-th rootof unity, then any closed subgroup of infinite index (respectively, any closed

subgroup of arbitrary index) HG(p)k is a free pro-p group.

Proof. Assertion (i) follows from the argument applied to verify Proposition 1.5,(i). To verify assertion (ii), it suffices to show that the cohomology module

Hj

(J, M) = limk Hj

(Gk

, M)

[where J def

= Ker(Gk G(p)k ); k

ranges over the finite Galois extensions ofk suchthat [k : k] is a power of p; Gk Gk is the open subgroup determined by k

]vanishesfor j 1. By devissage, we may assume that M = Fp with the trivialGk-action. Since the cohomological dimension of Gk is equal to 2 [cf. [NSW],Theorem 7.1.8, (i)], it suffices to consider the cases j = 1, 2. For j = 2, sinceH2(Gk ,Fp) = Fp [cf. [NSW], Theorem 7.1.8, (ii); our hypothesis thatk containsa primitivep-th root of unity], it suffices, by the well-known functorial behaviorofH2(Gk ,Fp) [cf. [NSW], Corollary 7.1.4], to observe that k

always admits a cyclic

Galois extension of degree p[arising, for instance, from an extension of the residuefield of k]. On the other hand, for j = 1, the desired vanishing is a tautology,

in light of the definition of the quotient Gk G(p)k . This completes the proof of

assertion (ii).


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Finally, we consider assertion (iii). Ifk does notcontain a primitivep-th root

of unity, then G(p)k itself is a free pro-p group [cf. [NSW], Theorem 7.5.8, (i)], so

any closed subgroup H G(p)k is also free pro-p [cf., e.g., [RZ], Corollary 7.7.5].

Thus, let us assume that k contains a primitive p-th root of unity, so we mayapply the isomorphism of assertion (ii). In particular, if J G

(p)k is an open

subgroup such that HJ, and kJk is the subfield determined by J, then oneverifies immediately that the quotientGkJ Jmay be identified with the quotient

GkJ G(p)kJ

, so we obtain an isomorphism H2(J,Fp) H2(GkJ,Fp) [where Fp is

equipped with the trivial Galois action]. Thus, to complete the proof thatH isfreepro-p, it suffices [by a well-known cohomological criterion for free pro-p groupscf., e.g., [RZ], Theorem 7.7.4] to show that the cohomology module

H2(H, Fp)=lim

kJH2(GkJ,Fp)

[where Fp is equipped with the trivial Galois action; kJ ranges over the finite

extensions ofk arising from open subgroups JG(p)k such thatHJ] vanishes.

As in the proof of assertion (ii), this vanishing follows from the well-knownfunctorialbehaviorofH2(GkJ,Fp), together with the observation that, by our assumption that

H is of infinite index in G(p)k , kJalways admits an extension of degree p arising

from an open subgroup ofJ [where JG(p)k corresponds to kJ] containing H.

Theorem 1.7. (Slimness and Elasticity of Arithmetic Galois Groups)Letk be asolvably closed Galois extension of a fieldk; writeGk def= Gal(k/k).Then:

(i) Ifk is anFF, thenGk=Z isneither elasticnor slim.(ii) If k is an MLF of residue characteristic p, then Gk, as well as any

almost pro-p-maximal quotient Gk Q ofGk, iselastic andslim.

(iii) Ifk is anNF, thenGk isvery elastic andslim.

Proof. Assertion (i) is immediate from the definitions; assertion (iii) is the contentof [Mzk11], Corollary 2.2; [Mzk11], Theorem 2.4. The slimnessportion of assertion(ii) for Gk is shown, for instance, in [Mzk6], Theorem 1.1.1, (ii) [via the sameargument as the argument applied to prove Proposition 1.5, (i); Proposition 1.6,(i)]; theslimnessportion of assertion (ii) for Qis precisely the content of Proposition1.6, (i). Write pfor the residue characteristicofk.

To show the elasticityportion of assertion (ii) for Q, let N Hbe a closednormal subgroup of infinite index of an open subgroup H Q such that N istopologically generated by r elements, where r 1 is an integer. Then it sufficesto show that N is trivial. Since Q has already been shown to be slim [hence hasno nontrivial finite normal closed subgroups cf. 0], we may always replace kby a finite extension of k. In particular, we may assume that H = Q, and thatQ is maximal pro-p. Since [Q : N] is infinite, it follows that there exists an opensubgroup J Q, corresponding to a subfield kJ k, such that N J, and


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[kJ : Qp] r+ 1. Here, we recall from our discussion oflocal class field theory inthe proof of Proposition 1.5 that dimQp(J

ab Qp) = [kJ : Qp] + 1 ( r+ 2). Inparticular, we conclude that N is necessarily a subgroup ofinfinite indexof some

topologically finitely generated closed subgroup P Jsuch that [J :P] is infinite.[For instance, one may take Pto be the subgroup ofJtopologically generated byN, together with an element ofJthat maps to a non-torsion element of the quotientofJab by the image ofNab.] Thus, we conclude from Proposition 1.6, (iii), that Pis afree pro-pgroupwhich contains a topologically finitely generated closed normalsubgroup N P of infinite index. On the other hand, by [a rather easy specialcase of] thetheorem of Lubotzky-Melnikov-van den Dries[cf., e.g., [FJ], Proposition24.10.3; [MT], Theorem 1.5], this implies that N istrivial. This completes the proofof the elasticity portion of assertion (ii) for Q.

To show the elasticityportion of assertion (ii) for Gk, let N Hbe a closed

normal subgroup of infinite index of an open subgroup H Gk such that N istopologically generated by r elements, where r 1 is an integer. Then it sufficesto show that N is trivial. As in the proof of the elasticity of Q, we may assumethat H = Gk; also, since [Gk : N] is infinite, by passing to a finite extension ofk corresponding to an open subgroup of Gk containing N, we may assume that[k : Qp] r. But this implies that the image ofN in G

abk Zp [which is of rank

[k : Qp] + 1 r+ 1] is of infinite index, hence that the image ofN in any almostpro-p-maximal quotientGk Qis of infinite index. Thus, by the elasticity of Q,we conclude that such images are trivial. Since, moreover, the natural surjection

Gk

limQ Q

[whereQranges over the almost pro-p-maximal quotients ofGk] is [by the definitionof the term almost pro-p-maximal quotient] an isomorphism, this is enough toconclude that N is trivial, as desired.


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Section 2: Semi-absolute Anabelian Geometry

In the present 2, we consider the problem ofcharacterizing group-theoretically

the quotient morphism to the Galois group of the base fieldof the arithmetic fun-damental group of a variety. In particular, the theory of the present 2 refines thetheory of [Mzk6], Lemma 1.1.4, in two respects: We extend this theory to the case ofquite general varieties of arbitrary dimension[cf. Corollary 2.8], and, in the case ofhyperbolic orbicurves, we give agroup-theoretic versionof the numerical criterionof [Mzk6], Lemma 1.1.4, via the theory ofmaximal pro-RTF-quotientsdeveloped in1 [cf. Corollary 2.12]. The theory of the present 2 depends on the general theoryof Albanese varieties, which we review in the Appendix, for the convenience of thereader.

Suppose that:

(1) k is a perfect field, k an algebraic closureofk,k k a solvably closedGalois extensionofk, and Gk

def= Gal(k/k).

(2) X Spec(k) is a geometrically connected, smooth, separated algebraicstack of finite typeover k.

(3) Y X is aconnected finite etale Galois coveringwhich is a [necessarilyseparated, smooth, and of finite type overk]k-schemesuch that Gal(Y /X)acts freelyon some nonempty open subscheme ofY [so X is generically

scheme-like cf. 0].(4) Y Y is an open immersion into a connected properk-schemeY such

thatYis the underlying scheme of a log scheme Ylog

that islog smoothoverk[where we regard Spec(k) as equipped with the trivial log structure], andthe image ofY in Y coincides with the interior[cf. 0] of the log scheme

Ylog

.

Thus, it follows from the log purity theorem [which is exposed, for instance, in[Mzk4] as Theorem B] that the condition that a finite etale covering Z Y

be tamely ramifiedover the height one primes ofY is equivalent to the conditionthat the normalization Z ofY in Z determine a log etale morphismZ

log Y

log

[whose underlying morphism of schemes is Z Y]; in particular, one concludesimmediately that the condition that Z Y be tamely ramified over the heightone primes ofY is independentof the choice of log smooth log compactification

Ylog

for Y. Thus, one verifies immediately [by considering the various Gal(Y /X)-

conjugatesof the log compactification Ylog

] that the finite etale coverings ofXwhose pull-backs toY aretamely ramifiedover [the height one primes of] Y form aGalois category, whose associated profinite group [relative to an appropriate choiceof basepoint for X] we denote by tame1 (X, Y), or simply

tame1 (X)

when Y X is fixed. In particular, if we use the subscript k to denote base-change from k to k, then by choosing a connected component of Yk, we obtain


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a subgroup tame1 (Xk) tame1 (X) which fits into a natural exact sequence 1

tame1 (Xk) tame1 (X) Gal(k/k) 1.

Next, let Primes be a set of prime numbers; tame1

(Xk

) X an al-most pro--maximal quotient of tame1 (Xk) whose kernel is normal in

tame1 (X),

hence determines a quotient tame1 (X) X ; we also assume that the quotienttame1 (X) Gal(Y /X) admits a factorization

tame1 (X) X Gal(Y /X),

and that the kernel of the resulting homomorphism X Gal(Y /X) is pro-.Thus, Ker(X Gal(Y /X)) may be identified with the maximal pro- quotientof Ker(tame1 (Xk) Gal(Y /X)); we obtain a natural exact sequence

1 X X Gal(k/k) 1

which may be thought of as an extensionof the profinite group Gal(k/k).

Definition 2.1.

(i) We shall refer to any profinite group which is isomorphic to the profinitegroup X constructed in the above discussion for some choice of data (k , X , Y Y , ) as a profinite groupof [almost pro-] GFG-type[where GFG is to be under-stood as an abbreviation forgeometric fundamental group]. In this situation, weshall refer to any surjection tame1 (Xk) obtained by composing the surjection

tame1 (Xk) X with an isomorphism X as a scheme-theoretic envelope

for ; we shall refer to (k , X , Y Y , ) as a collection of construction data for

. [Thus, given a profinite group of GFG-type, there are, in general, many possiblechoicesof construction data for the profinite group.]

(ii) We shall refer to any extension 1 G 1 of profinite groupswhich is isomorphic to the extension 1 X X Gal(k/k) 1 constructedin the above discussion for some choice of data (k , X , Y Y , ) as an extensionof [geometrically almost pro-] AFG-type[where AFG is to be understood as anabbreviation for arithmetic fundamental group]. In this situation, we shall referto any surjection tame1 (X) (respectively, any surjection

tame1 (Xk) ; any

isomorphism Gal(k/k) G) obtained by composing the surjection tame1 (X) X

(respectively, the surjection tame1 (Xk) X ; the identity Gal(k/k) = Gal(k/k))

with an isomorphism X (respectively, X

; Gal(k/k)

G) arising

from an isomorphism of the extensions 1 G 1, 1 X X Gal(k/k) 1 as a scheme-theoretic envelope for (respectively, ; G); we shallrefer to (k , X , Y Y , ) as a collection of construction data for this extension.[Thus, given an extension of AFG-type, there are, in general, many possible choicesof construction data for the extension.]

(iii) Let 1 G 1 be an extension of AFG-type; N G

the inverse image of the kernel of the quotient Gal(k/k) Gk relative to some

scheme-theoretic envelope Gal(k/k) G. Suppose further that is slim, and

that the outer action ofN on [arising from the extension structure] is trivial.Thus, every element ofNG lifts to a unique element of thatcommuteswith. In particular, N lifts to a closed normal subgroup N . We shall referto any extension 1 G 1 of profinite groups which is isomorphic


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to an extension of the form 1 /N G/N 1 just constructed asan extension of [geometrically almost pro-] GSAFG-type[where GSAFG is tobe understood as an abbreviation for geometrically slim arithmetic fundamental

group]. In this situation, we shall refer to any surjectiontame1 (X) (respec-

tively, tame1 (Xk) ; Gal(k/k) G) obtained by composing a scheme-theoretic

envelope tame1 (X) (respectively, tame1 (Xk)

; Gal(k/k) G) with

the surjection (respectively, ; G G) in the above discus-sion as a scheme-theoretic envelope for (respectively, ; G); we shall refer to

(k, k , X , Y Y , ) as a collection of construction data for this extension. [Thus,given an extension of GSAFG-type, there are, in general, many possible choicesofconstruction data for the extension.]

(iv) Given construction data(k , X , Y Y , ) or (k, k , X , Y Y , ) asin (i), (ii), (iii), we shall refer to k as the construction data field, to X as theconstruction data base-stack[or base-scheme, ifXis a scheme], to Y as the con-struction data covering, to Y as the construction data covering compactification,and to as the construction data prime set. Also, we shall refer to a portion

of the construction data (k , X , Y Y , ) or (k, k , X , Y Y , ) as in (i),(ii), (iii), as partial construction data. If every prime dividing the order of a finitequotient group of is invertible in k, then we shall refer to the construction datain question as base-prime.

The following result is well-known, but we give the proof below for lack of an

appropriate reference in the case where [in the notation of the above discussion] Xis not necessarily proper.

Proposition 2.2. (Topological Finite Generation) Any profinite group of GFG-type istopologically finitely generated.

Proof. Write (k , X , Y Y , ) for a choice of construction data for . Sincea profinite fundamental group is topologically finitely generated if and only if itadmits an open subgroup that is topologically finitely generated, we may assumethat X = Y; moreover, by applying de Jongs theory ofalterations [as reviewed,

for instance, in Lemma A.10 of the Appendix], we may assume that Y isprojectiveand k-smooth, and that D

def= Y \ Y is a divisor with normal crossingson Y. Since

we are only concerned with , we may assume that k isalgebraically closed, hence,in particular, infinite. Now suppose that dim(Y) 2. Then since Y is smoothand projective [over k], it follows that there exists a connected, k-smooth closedsubschemeC Yobtained by intersectingY with a sufficiently general hyperplanesection H such that D

H forms a divisor with normal crossings on C. Write

C def

= C

Y (= ). Now ifZ Y is any connected finite etale covering that istamely ramifiedover D, then write ZY for the normalizationofY in Z. Thus,since Z is tamely ramfiedover D so, by the log puritytheorem reviewed above,

one may apply the well-known theory of log etalemorphisms to describe the localstructure of Z Y and D intersects C transversely, it follows immediately

that ZCdef= ZY C is normal. On the other hand, since the closed subscheme

ZC Z arises as the zero locus of a nonzero section of an ample line bundle


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on the normalscheme Z, it thus follows [cf., [SGA2], XI, 3.11; [SGA2], XII, 2.4]that ZC is connected, hence [since ZC is normal] irreducible. But this implies

that ZCdef= ZY C = ZC

Y is connected. Moreover, this connectedness ofZC

for arbitrary choices of the covering Z Y implies that the natural morphismtame1 (C)

tame1 (Y) is surjective. Thus, by induction on dim(Y), it suffices to

prove Proposition 2.2 in the case where Y is a curve. But in this case, [as is well-known] Proposition 2.2 follows by deformingY Y to a curve in characteristiczero, in which case the desired topological finite generation follows from the well-known structure of thetopological fundamental group of a Riemann surface of finitetype.

Proposition 2.3. (Slimness and Elasticity for Hyperbolic Orbicurves)

(i) Let be a profinite group of GFG-type that admits partial constructiondata (k,X, ) [consisting of the construction data field, construction data base-stack, and construction data prime set] such thatX is ahyperbolic orbicurve[cf. 0], and contains aprime invertible in k. Then isslim andelastic.

(ii) Let 1 G 1 be an extension of GSAFG-type thatadmits partial construction data(k,X, )[consisting of the construction data field,construction data base-stack, and construction data prime set] such that X is ahyperbolic orbicurve, = , and k is either an MLF or an NF. Then isslim, butnot elastic.

Proof. Assertion (i) is the easily verifiedgeneralization to orbicurves over fieldsof arbitrary characteristic of [MT], Proposition 1.4; [MT], Theorem 1.5 [cf. alsoProposition 1.3, (i)]. The slimnessportion of assertion (ii) follows immediatelyfrom the slimness portion of assertion (i), together with the slimness portion ofTheorem 1.7, (ii), (iii); the fact that is not elastic follows from the existence ofthe nontrivial, topologically finitely generated [cf. Proposition 2.2], closed, normal,infinite index subgroup .

Definition 2.4. For i= 1, 2, let

1 i i Gi 1

be an extension which is eitherof AFG-typeor of GSAFG-type. Suppose that

: 1 2

is a continuous homomorphism of profinite groups. Then:

(i) We shall say that isabsoluteif is open [i.e., has open image].

(ii) We shall say that is semi-absolute(respectively, pre-semi-absolute) ifis absolute, and, moreover, the image of(1) in G2 is trivial(respectively, eithertrivialor of infinite indexin G2).


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(iii) We shall say thatisstrictly semi-absolute (respectively,pre-strictly semi-absolute) if is semi-absolute, and, moreover, the subgroup (1) 2 is open(respectively, eitheropenor nontrivial).

Proposition 2.5. (First Properties of Absolute Homomorphisms) Fori= 1, 2, let

1 i i Gi 1

be an extension which is either of AFG-type or of GSAFG-type; (ki, Xi, i)partial construction datafori Gi [consisting of the construction data field,construction data base-stack, and construction data prime set]. Suppose that

: 1 2

is acontinuous homomorphism of profinite groups. Then:

(i) The following implications hold:

strictly semi-absolute = pre-strictly semi-absolute = semi-absolute

= pre-semi-absolute = absolute.

(ii) Suppose that k2 is an NF. Then semi-absolute pre-semi-absolute absolute.

(iii) Suppose that k2 is an MLF. Then semi-absolute pre-semi-absolute.

(iv) Suppose that k1 either an FF or an MLF; that X2 is a hyperbolicorbicurve; and that2 is ofcardinality> 1. Then pre-strictly semi-absolute semi-absolute.

(v) Suppose that X2 is a hyperbolic orbicurve, and that 2 contains aprime invertible in k2. Then strictly semi-absolutepre-strictly semi-absolute.

Proof. Assertion (i) follows immediately from the definitions. Since 1 is topo-logically finitely generated [cf. Proposition 2.2], assertion (ii) (respectively, (iii))follows immediately, in light of assertion (i), from the fact that G2 is very elastic[cf. Theorem 1.7, (iii)] (respectively, elastic [cf. Theorem 1.7, (ii)]). To verifyassertion (iv), it suffices, in light of assertion (i), to consider the case where issemi-absolute, but not pre-strictly semi-absolute. Then since 2 is elastic [cf. thehypothesis on 2; Proposition 2.3, (i)], and 1 istopologically finitely generated[cf.Proposition 2.2], it follows that the subgroup (1) 2 is either openor trivial.Since is not pre-strictly semi-absolute, we thus conclude that (1) ={1}, so induces an open homomorphismG1 2. That is to say, every sufficiently smallopen subgroup 2 2 admits a surjectionH1

2 for some closed subgroup

H1 G1. On the other hand, since X2 is a hyperbolic orbicurve, and 2 is ofcardinality > 1, it follows [e.g., from the well-known structure of topological fun-damental groups of hyperbolic Riemann surfaces of finite type] that we may take


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2 such that 2 admits quotients

2 F

, 2 F, where F (respectively,

F) is anonabelianfree pro-p (respectively, pro-p) group, fordistinctp, p 2.But this contradicts the well-known structure of G1, when k1 is either an FF or

an MLF i.e., the fact that G1, hence also H1, may be written as an extensionof a meta-abelian group by a pro-p subgroup, for some prime p. [Here, we recallthat this fact is immediate ifk1 is an FF, in which case G1 is abelian, and follows,for instance, from [NSW], Theorem 7.5.2; [NSW], Corollary 7.5.6, (i), when k1 is aMLF.] Assertion (v) follows immediately from the elasticityof 2 [cf. Proposition2.3, (i)], together with the topological finite generationof 1 [cf. Proposition 2.2].

Theorem 2.6. (Field Types and Group-theoreticity)Let

1 G 1

be an extension which is either of AFG-type or of GSAFG-type; (k,X, )partial construction data [consisting of the construction data field, constructiondata base-stack, and construction data prime set] for G. Suppose further thatk is either anFF, anMLF, or anNF, and that every prime is invertibleink. IfHis a profinite group, j {1, 2}, and l Primes, write

jl (H)def= dimQl(H

j(H,Ql)) N

{}

jl ()def= supJ {

jl (J)} N

{}

j()def= {l | jl () 3 j} Primes

[whereJranges over the open subgroups of]; also, we set

(H)def= sup

p,pPrimes{1p(H)

1p(H)} Z

{}

whenever1l(H)< , l Primes. Then:

(i) Suppose thatk is anFF. Then istopologically finitely generated;

the natural surjectionsab-t Gab-t; G Gab-t

areisomorphisms. In particular, the kernel of the quotient G may be char-acterized [group-theoretically] as the kernel of the quotient ab-t [cf.[Tama1], Proposition 3.3, (ii), in the case of curves]. Moreover, for every opensubgroupH, and every prime number l, 1l(H) = 1.

(ii) Suppose thatk is anMLF of residue characteristicp. Then istopologi-cally finitely generated; in particular, for every open subgroup H, and everyprime numberl,1l(H)isfinite. Moreover,

1l(G) = 1ifl =p,

1p(G) = [k: Qp]+ 1;

the quantity1l()

1l(G)

is = 0 if l / , and is independent of l if l . Finally, 1p() = ; in

particular, thecardinality of1() is always1.


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(iii) Letk be as in (ii). Then2() . If, moreover, the cardinality of1()is2, then2() = .

(iv) Letk be as in (ii). Then everyalmost pro-omissivetopologically finitelygenerated closed normal subgroup of is contained in . If, moreover, =Primes, then the kernel of the quotient G may be characterized [group-theoretically] as themaximal almost pro-omissive topologically finitely gen-erated closed normal subgroup of.

(v) Letk be as in (ii). If2()=Primes, then write

for themaximal almost pro-omissive topologically finitely generated closed nor-

mal subgroup of , whenever a unique such maximal subgroup exists; if

2

() =Primes, or there does not exist a unique such maximal subgroup, set

def= {1} .

Then()

def= (/) = [k: Qp]

[cf. the finiteness portion of (ii)]. In particular, the kernel of the quotient Gmay be characterized [group-theoretically since 2(), (), ()are group-theoretic] as the intersection of the open subgroups H such that(H)/() = [ :H].

(vi) Suppose thatk is anNF. Then the natural surjectionab-t Gab-t is an

isomorphism. The kernel of the quotient G may be characterized [group-theoretically] as the maximal topologically finitely generated closed normal sub-group of. In particular, isnot topologically finitely generated.

Proof. Write X A for the Albanese morphism associated to X. [We refer tothe Appendix for a review of the theory of Albanese varieties cf., especially,Corollary A.11, Remark A.11.2.] Thus, A is a torsorover a semi-abelian varietyover k such that the morphism X A induces an isomorphism

ab-t Zl Tl(A)

onto thel-adic Tate moduleTl(A) ofA for alll . Note, moreover, that forl ,the quotient of determined by the imageof in the pro-l completion of ab-t

factors throughthe quotient

ab-t Zl Tl(A) Tl(A)/G

where we use the notation /G to denote the maximal torsion-free quotientonwhichG acts trivially.

Next, whenever k is an MLF, let us write, for l ,

ab-t ab-t ZlTl(A) Rl

def= R Zl Ql

def= Q Zl

for the pro-l portion of the quotientsT(A) R Qof Lemma 2.7, (i), (ii), below[in which we take k to be k and B to be the semi-abelian variety over which


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Ais a torsor]. Here, we observe that Ql is simply the quotient Tl(A)/Gconsideredabove. Thus, the Zl-ranks ofRl, Ql are independentofl .

The topological finite generation portion of assertion (i) follows immediately

from the fact that G=Z, together with the topological finite generation of [cf.Proposition 2.2]. The remainder of assertion (i) follows immediately from the factthat Tl(A)/G= 0 [a consequence of the Riemann hypothesis for abelian varietiesover finite fields cf., e.g., [Mumf], p. 206]. In a similar vein, assertion (vi)follows immediately from the fact that Tl(A)/G = 0 [again a consequence of theRiemann hypothesis for abelian varieties over finite fields], together with the factthat G is very elastic [cf. Theorem 1.7, (iii)].

To verify assertion (ii), let us first observe that the topological finite generationof follows from that of [cf. Proposition 2.2], together with that ofG [cf. [NSW],

Theorem 7.5.10]. Next, let us recall the well-known fact that

1l(G) = 1 ifl=p, 1p(G) = [k: Qp] + 1

[cf. our discussion oflocal class field theoryin the proofs of Proposition 1.5; Theorem1.7, (ii)]; in particular, (G) = [k: Qp]. Moreover, the existence of a rational pointofAover some finite extension ofk [which determines a Galois section of the etalefundamental group ofA over some open subgroup ofG] implies that

1l() =1l(G) + dimQl(Ql Ql)

[where we recall that dimQl(Ql Ql) is independentofl] for l , 1l() =

1l(G)

forl /. Thus, by considering extensions ofk of arbitrarily large degree, we obtainthat 1p() =. This completes the proof of assertion (ii).

Next, we consider assertion (iii). First, let us consider the E2-term of theLeray spectral sequence of the group extension 1 G 1. Since G isofcohomological dimension2 [cf., e.g., [NSW], Theorem 7.1.8, (i)], and 2l(G) = 0for all l Primes[cf., e.g., [NSW], Theorem 7.2.6], the spectral sequence yields anequality2l() = 0 if l /, and a pair of injections

H1

(G, Hom(Rl,Ql)) H1

(G, Hom(ab-t

,Ql)) H2

(,Ql)

if l [cf. Lemma 2.7, (iii), below]. By applying the analogue of this conclusionfor an arbitrary open subgroup H , we thus obtain that 2l(H) = 0 if l / ,i.e., that 2l () = 0 if l / ; this already implies that if l / , then l /

2(),i.e., that 2() . If the cardinality of 1() is 2, then there exists someopen subgroup H and some l Primes such that 1l(H) 2, l = p. Nowwe may assume without loss of generality that H acts trivially on the quotientR; also to simplify notation, we may replace by H and assume that H = .Then [since 1l(G) = 1, by assertion (ii)] the fact that

1l() 2 implies that

l , and dimQl(Rl Ql) 1 [cf. our computation in the proof of assertion (ii)].But this implies that for any l , we have dimQl (Rl Ql) 1, hence thatH1(G, Hom(Rl ,Ql)) = H

1(G,Ql) Hom(Rl ,Ql) = 0. Thus, by the injectionsdiscussed above, we conclude that 2l()

2l() 1, so l

2(). This completesthe proof of assertion (iii).


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Assertion (iv) follows immediately from the existence of a surjectionG Z[cf., e.g., Proposition 1.5, (ii)], together with the elasticityofG [cf. Theorem 1.7,(ii)], and the topological finite generationof [cf. Proposition 2.2].

Next, we consider assertion (v). First, let us observe that whenever =Primes, it follows from assertion (ii) that () =(G) = [k: Qp].

Now we consider the case 2() = Primes. In this case, = {1} [by def-inition], and 2( ) = = Primes [by assertion (iii)]. Thus, we obtain that() = (/ ) = [k : Qp], as desired [cf. [Mzk6], Lemma 1.1.4, (ii)]. Next,we consider the case 1() = {p} [i.e., 1() is of cardinality 2 cf. as-sertion (ii)], 2() = Primes. In this case, by assertion (iii), we conclude that =2()=Primes. Thus, by assertion (iv), = , so(/) =(G) = [k: Qp],as desired.

Finally, we consider the case 1() = {p} [i.e., 1() is of cardinality one],2()= Primes. If = Primes, then it follows from the definition of , togetherwith assertion (iv), that = , hence that (/) =(G) = [k : Qp], as desired.If, on the other hand, = Primes, then since 1() = {p}, it follows [cf. thecomputation in the proof of assertion (ii)] that dimQl(Ql Ql) = 0 for all primesl = p, hence that dimQp(Qp Qp) = 0; but this implies that

1l() =

1l(G) for

all l Primes. Now since [by assertion (iv)], it follows that 1l() 1l(/)

1l(G) for all l Primes, so we obtain that

1l() =

1l(/) =

1l(G)

for all l Primes. But this implies that () = (/) = (G) = [k : Qp], asdesired. This completes the proof of assertion (v).

Remark 2.6.1. When [in the notation of Theorem 2.6] X is a smooth propervariety, the portion of Theorem 2.6, (ii), concerning 1l()

1l(G) is essentially

equivalent to the main result of [Yoshi].

Lemma 2.7. (Combinatorial Quotients of Tate Modules)Suppose that

k is anMLF [so k= k]. LetB be asemi-abelian variety overk. Write

T(B)def= Hom(Q/Z, B(k))

for the Tate module ofB. Then:

(i) The maximal torsion-freequotient moduleT(B) Q ofT(B) on which

Gk actstrivially is a finitely generatedfreeZ-module.(ii) There exists aquotient Gk-module T(B) R such that the following

properties hold: (a) R is a finitely generated freeZ-module; (b) the action ofGkonR factors through a finite quotient; (c) no nonzero torsion-free subquotientSof

theGk-moduleN def= Ker(T(B) R)satisfies the property that the resulting action

ofGk onSfactors through a finite quotient.(iii) IfR is as in (ii), then the natural map

H1(Gk, Hom(R,Z)) H1(Gk, Hom(T(B),Z))


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isinjective.

Proof. Assertion (i) is literally the content of [Mzk6], Lemma 1.1.5. Assertion

(ii) follows immediately from the proof of [Mzk6], Lemma 1.1.5 [more precisely, thecombinatorialquotient Tcom of loc. cit.]. Assertion (iii) follows by consideringthe long exact cohomology sequence associated to the short exact sequence 0 Hom(R,Z) Hom(T(B),Z) Hom(N,Z) 0, since the fact that N has nononzero torsion-free subquotients on whichGkacts through a finite quotient implies

that H0(Gk, Hom(N,Z)) = 0.

Corollary 2.8. (Field Types and Absolute Homomorphisms)Fori = 1, 2,let 1 i i Gi 1, ki, Xi, i, : 1 2 be as in Proposition 2.5.Suppose further thatki is either anFF, anMLF, or anNF, and that every primei is invertible inki. Then:

(i) Suppose further that is absolute. Then thefield type of k1 is [cf.0] the field type ofk2. If, moreover, it holds either that bothk1 andk2 areFFsor that bothk1 andk2 areNFs, then issemi-absolute, i.e., (1) 2.

(ii) Suppose further that is anisomorphism. Then thefield types ofk1,k2 coincide, and isstrictly semi-absolute, i.e., (1) = 2. If, moreover,fori= 1, 2, ki is anMLF of residue characteristicpi, thenp1=p2.

Proof. Assertion (i) concerning the inequality follows immediately from thetopological finite generation portions of Theorem 2.6, (i), (ii), (vi), together withthe estimates of 1l(),

1l () in Theorem 2.6, (i), (ii). The final portion of

assertion follows, in the case of FFs, from Theorem 2.6, (i), and, in the case ofNFs, from Proposition 2.5, (ii). Next, we consider assertion (ii). The fact thatthe field typesofk1, k2 coincide follows from assertion (i) applied to ,

1. Toverify that is strictly semi-absolute, let us first observe that every semi-absoluteisomorphism whose inverse is also semi-absolute is necessarily strictly semi-absolute.Thus, since the inverse to satisfies the same hypotheses as , to complete theproof of Corollary 2.8, it suffices to verify that is semi-absolute. If k1, k2 are

FFs (respectively, MLFs; NFs), then this follows immediately from the group-theoretic characterizationsof i Gi in Theorem 2.6, (i) (respectively, Theorem2.6, (v); Theorem 2.6, (vi)). Finally, if, for i = 1, 2, ki is an MLF of residue

characteristic pi, then since induces an isomorphism G1 G2, the fact that

p1 =p2 follows, for instance, from [Mzk6], Proposition 1.2.1, (i).

Remark 2.8.1. In the situation of Corollary 2.8, suppose further that k2 is anMLF of residue characteristicp2; thatX2 is ahyperbolic orbicurve; that 2 {p2}[cf. Proposition 2.5, (iv)]; and that if 2 =, thenk1 is an NF. Then it is not clearto the author at the time of writing [but of interest in the context of the theory of

the present2!] whether or not there exists a continuous surjective homomorphism

G1 2

[in which case, by Corollary 2.8, (i), k1 is either an NF or an MLF].


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The general theory discussed so far for arbitraryXbecomes substantially sim-pler and more explicit, when X is a hyperbolic orbicurve.

Definition 2.9. LetG be aprofinite group. Then we shall refer to as anaug-freedecomposition of G any pair of closed subgroups H1, H2 G that determine anisomorphism of profinite groups

H1 H2G

such that H1 is a slim, topologically finitely generated, augmented pro-prime [cf.Definition 1.1, (iii)] profinite group, and H2 is either trivialor a nonabelian pro--solvable free group for some set Primes of cardinality 2. In this situation,we shall refer to H1 as the augmented subgroup of this aug-free decomposition and

to H2 as the free subgroup of this aug-free decomposition. IfG admits an aug-freedecomposition, then we shall say that G is of aug-free type. IfG is of aug-free type,withnontrivialfree subgroup, then we shall say that Gisof strictly aug-free type.

Proposition 2.10. (First Properties of Aug-free Decompositions) Let

H1 H2 G

be anaug-free decomposition of a profinite group G, in which H1 is the aug-mented subgroup, andH2 is the free subgroup. Then:

(i) Let J be a topologically finitely generated, augmented pro-primegroup; : J G a continuous homomorphism of profinite groups such that(J)isnormal in some open subgroup ofG. Then(J) H1.

(ii) Aug-free decompositions areunique i.e., ifJ1 J2 G is any aug-

free decomposition ofG, in whichJ1 is the augmented subgroup, andJ2 is the freesubgroup, thenJ1 =H1, J2=H2.

Proof. First, we consider assertion (i). Suppose that(J) is notcontained in H1.

Then the imageIH2of(J) via the projection toH2is anontrivial,topologicallyfinitely generatedclosed subgroup which isnormalin an open subgroup ofH2. SinceH2 is elastic [cf. [MT], Theorem 1.5], it follows that I is open in H2, hence thatI is a nonabelian pro--solvable free group for some set Primes ofcardinality2. On the other hand, since Iis a quotient of the augmented pro-primegroup J,it follows that there exists a p Primes such that the maximal pro-(=p) quotientofI is abelian. But this implies that {p}, a contradiction. Next, we considerassertion (ii). By assertion (i), J1 H1, H1 J1. Thus, H1 = J1. Now sinceH1 = J1 is slim, it follows that the centralizer ZH1(G) (respectively, ZJ1(G)) isequal to H2 (respectively, J2), so H2 =J2, as desired.

Theorem 2.11. (Maximal Pro-RTF-quotients for Hyperbolic Orbi-curves)Let

1 G 1


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be an extensionof AFG-type; (k,X, ) partial construction data [consistingof the construction data field, construction data base-stack, and construction dataprime set] for G. Suppose thatk is anMLF of residue characteristicp; X is

a hyperbolic orbicurve; =. For l Primes, write

[l]

for themaximal almost pro-l topologically finitely generated closed normal sub-group of, whenever a unique such maximal subgroup exists; if there does not exist

a unique such maximal subgroup, then set[l]def= {1}.

In the following, we shall use a subscript G to denote the quotient of aclosed subgroup of determined by the quotient G; we shall use the super-script RTF to denote the maximal pro-RTF-quotient and the superscripts

RTF-aug, RTF-free to denote theaugmentedandfreesubgroups of the max-imal pro-RTF-quotient whenever this maximal pro-RTF-quotient is of aug-freetype. Then:

(i) Suppose that [l] = {1} for some l Primes. Then [l] = , = {l};[l] ={1} for all l Primes such that l =l.

(ii) Suppose that[l] ={1}for alll Primes. Thenis ofcardinality 2.Moreover, for every open subgroup J , there exists an open subgroup H Jwhich ischaracteristic as a subgroup of such thatHRTF isof aug-free type.In particular, [cf. Proposition 2.10, (ii)] the subquotientsHRTF-aug, HRTF-free of

arecharacteristic.

(iii) Suppose that [l] = {1} for all l Primes. Suppose, moreover, thatH is an open subgroup that corresponds to afinite etale covering ZX,where Z is a hyperbolic curve, defined over a finite extension kZ of k suchthat Z has stable reduction [cf. 0] over the ring of integers OkZ of kZ; thatZ(kZ)=; that thedual graph Zof the geometric special fiber of the resultingmodel [cf. 0] overOkZ has eithertrivial ornonabelian topological fundamentalgroup; and that the Galois action of G on Z is trivial. Thus, the finite Galoiscoverings of the graph Z of degree a product of primes determine a pro-combinatorial quotientH com

H ; writecom

H com-sol

H for themaximal

pro-solvable quotient ofcomH . Then the quotient

H HRTFG com-solH

may be identified with themaximal pro-RTF-quotient H HRTF ofH; more-over, this product decomposition determines anaug-free decompositionofHRTF.Finally, for any open subgroup J, there exists an open subgroup HJ whichischaracteristic as a subgroup of and, moreover, satisfies the above hypotheseson H.

(iv) Suppose that [l] = {1} for all l Primes. Let H J be opensubgroups of such thatHRTF, JRTF are of aug-free type. Then we have iso-morphisms

JRTF-aug JRTFG ; J

RTF-free Ker(JRTF JRTFG )


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[arising from the natural morphisms involved]; the open homomorphismHRTF JRTF induced by mapsHRTF-aug (respectively,HRTF-free) onto an open subgroupofJRTF-aug (respectively,JRTF-free).

Proof. Since is elastic [cf. Proposition 2.3, (i)], every nontrivial topologicallyfinitely generated closed normal subgroup of is open, hence almost pro- for Primes if and only if . Also, let us observe that by Theorem 2.6, (iv),[l] for all l Primes. Thus, if [l]={1}for somel Primes, then it followsthat ={l}, [l] = , and that [l] is finite, hence trivial[since is slim cf.Proposition 2.3, (i)] for primes l = l. Also, we observe that if is of cardinalityone, i.e., = {l} for some l Primes, then = [l] = {1} [cf. Theorem 2.6,(iv)]. This completes the proof of assertion (i), as well as of the portion of assertion(ii) concerning . Also, we observe that the remainder of assertion (ii) followsimmediately from assertion (iii).

Next, we consider assertion (iii). Suppose that H satisfies the hypotheses

given in the statement of assertion (iii); write Hdef=

H. Thus, one has the

quotientH comH , where comH is eithertrivialor a nonabelian pro- free group,

and is of cardinality 2 [cf. the portion of assertion (ii) concerning ]. WriteabH =

ab-tH R for the maximal pro- quotient of the quotient R of Lemma

2.7, (ii), associated to the Albanese variety ofZ.

Now I claim that the quotient H R coincideswith the quotient H (comH )

ab. First, let us observe that by the definition ofR [cf. Lemma 2.7, (ii)], itfollows that the quotient H (

comH )

ab factors through the quotient HR.

In particular, since, forl , the modulesRZl, (comH )ab Zlare Zl-free modulesof rankindependentofl [cf. Lemma 2.7, (ii); the fact that comH is pro- free],it suffices to show that these two ranks are equal, for some l . Moreover, letus observe that for the purpose of verifying this claim, we may enlarge . Thus,it suffices to show that the two ranks are equal for some l such that l = p.But then the claim follows immediately from the [well-known] fact that by theRiemann hypothesis for abelian varieties over finite fields [cf., e.g., [Mumf], p.206], all powers of the Frobenius element in the absolute Galois group of the residuefield ofk act with eigenvalues= 1 on the pro-l abelianizations of the fundamentalgroups of the geometric irreducible components of the smooth locus of the special

fiber of the stable model ofZoverOkZ . This completes the proof of the claim.Now let us writeH Hcom for the quotient ofHby Ker(H

comH ). Then

by applying the above claim to various open subgroups of H, we conclude thatthe quotient H HRTF factorsthrough the quotient HHcom [i.e., we have a

natural isomorphismHRTF (Hcom)RTF]. On the other hand, since Z(kZ) = ,

it follows that H HG, hence also Hcom

HG admits a sections: HG Hcom

whose image lies in the kernel of the quotient Hcom comH [cf. the proof of[Mzk3], Lemma 1.4]. In particular, we conclude that the conjugation action ofHGon comH

=Ker(Hcom HG) Hcom arising from s is trivial. Thus, s determines

a direct product decomposition

Hcom HG comH

hence a direct product decomposition HRTF (Hcom)RTF

HRTFG (

comH )

RTF.Moreover, since comH is either trivialor nonabelian pro- free, it follows immedi-ately that the quotient comH (

comH )

RTF may be identified with the quotient


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comH com-solH , where

com-solH is eithertrivialor nonabelian pro--solvable free.

Since HRTFG is slim, augmented pro-prime, and topologically finitely generated [cf.Proposition 1.5, (i), (ii); Theorem 2.6, (ii)], we thus conclude that we have obtained

an aug-free decompositionofHRTF

, as asserted in the statement of assertion (iii).Finally, given an open subgroup J , the existence of an open subgroup

H Jwhich satisfies the hypotheses on H in the statement of assertion (iii)follows immediately from well-known facts concerning stable curves over discretelyvalued fields [cf., e.g., the stable reduction theorem of [DM]; the fact that =,so that one may assume that Z is as large as one wishes by passing to admissiblecoverings]. The fact that one can choose Hto becharacteristicfollows immediatelyfrom the characteristic nature of [cf., e.g., Corollary 2.8, (ii)], together with thefact that , aretopologically finitely generated[cf., e.g. Proposition 2.2; Theorem2.6, (ii)]. This completes the proof of assertion (iii).

Finally, we consider assertion (iv). First, we observe that since the augmentedand free subgroups of any aug-free decomposition are slim[cf. Definition 2.9; [MT],Proposition 1.4], hence, in particular, do not contain any nontrivial closed normal fi-nite subgroups, we may alwaysreplaceHby an open subgroup ofHthat satisfies thesame hypotheses as H. In particular, we may assume that His an open subgroupH as in assertion (iii) [which exists, by assertion (iii)]. Then by Proposition 2.10,(i), the image ofHRTF-aug in JRTF is contained inJRTF-aug, so we obtain a mor-phism HRTF-aug JRTF-aug. By assertion (iii), HRTF-free = Ker(HRTF HRTFG ),and the natural morphismHRTF-aug HRTFG is anisomorphism. SinceHG JG,hence also HRTFG J

RTFG , is clearly an open homomorphism, we thus conclude

that the natural morphism HRTF-aug JRTFG , hence also the natural morphismJRTF-aug JRTFG , isopen. Thus, the image ofJ

RTF-free inJRTFG commuteswith anopen subgroup ofJRTFG [i.e., the image ofJ

RTF-aug in JRTFG ], so by theslimnessofJRTFG [cf. Proposition 1.5, (i)], we conclude thatJ

RTF-free Ker(JRTF JRTFG ).In particular, we obtain a surjection JRTF-aug JRTFG , hence anexact sequence

1 N JRTF-aug JRTFG 1

where we write N def

= Ker(JRTF-aug JRTFG ) JRTF-aug JRTF. Note,

moreover, that since JRTFG is an augmented pro-p group [cf. Proposition 1.5, (ii)]which admits a surjection JRTFG ZpZp [cf. the computation of

1p() in

Theorem 2.6, (ii)], it follows immediately that [the augmented pro-prime group]JRTF-aug is an augmented pro-pgroup whose augmentation factorsthrough JRTFG ;in particular, we conclude thatNispro-p. Also, we observe that since the compositeHRTF-free HRTFG J

RTFG is trivial, it follows that the projection under the

quotientJRTF JRTF-aug of the image ofHRTF-free in JRTF is contained inN.

Now I claimthat to complete the proof of assertion (iv), it suffices to verifythat N = {1} [or, equivalently, since JRTF-aug is slim, that N is finite]. Indeed,if N = {1}, then we obtain immediately the isomorphisms JRTF-aug

JRTFG ,

JRTF-free Ker(JRTF JRTFG ). Moreover, by the above discussion, ifN = {1},

then it follows that the image ofHRTF-free inJRTF iscontained inJRTF-free. Sincethe homomorphism HRTF JRTF is open, this implies that the open homomor-phism HRTF JRTF induced by maps HRTF-aug (respectively, HRTF-free) ontoan open subgroup ofJRTF-aug (respectively,JRTF-free), as desired. This completesthe proof of the claim.


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Next, let J J be an open subgroup that arises as the inverse image in Jof an [open] RTF-subgroup JG JG [so the notation JG does not lead to anycontradictions]. Then one verifies immediately from the definitions that any RTF-

subgroup ofJG (respectively, J) determines an RTF-subgroup ofJG (respectively,J). Thus, the natural morphisms

JRTFG JRTFG ; J

RTF JRTF

are injective. Moreover, the subgroups JRTF-aug

JRTF, JRTF-free ofJRTF clearly

determine an aug-free decomposition of JRTF. Thus, from the point of view ofverifying the finiteness of N, we may replace J by J [and Hby an appropriatesmaller open subgroup contained in Jand satisfying the hypotheses of the H of(iii)]. In particular, since by the

definitionof RTF and of the subgroup N!

there exists a Jsuch that N JRTF-aug hasnontrivial imagein (JRTF-aug)ab-t, wemay assume without loss of generality that Nhasnontrivial imagein (JRTF-aug)ab-t.Thus, we have

(1p(J)) 1p(J

RTF-aug)> 1p(JRTFG ) =

1p(JG)

[cf. the notation of Theorem 2.6], i.e., sJdef= 1p(J

RTF-aug) 1p(JRTFG ) > 0. By

Theorem 2.6, (ii), this already implies that p .

In a similar vein, let J J be an open subgroup that arises as the inverseimage in J of an [open] RTF-subgroup JRTF-free JRTF-free. Then one verifiesimmediately from the definitions that any RTF-subgroup ofJdetermines an RTF-subgroup ofJ. Thus, the natural morphismJRTF JRTF is injective, with imageequal toJRTF-aug JRTF-free. Moreover, the subgroupsJRTF-aug,JRTF-free ofJRTF

clearly determine an aug-free decomposition ofJRTF [so the notation JRTF-freedoes not lead to any contradictions]. Since [by the above discussion applied to

J instead ofJ] JRTF-free maps to the identity in JRTFG , we thus obtain a quotient

JRTF JRTF-aug =JRTF-aug JRTFG , hence a quotientJRTF

JRTF-aug JRTFGin which the image ofJ

is afinitenormal closed subgroup, hence trivial [since

JRTFG is slim cf. Proposition 1.5, (i)]. That is to say, the surjection J JRTF-aug JRTFG to the pro-RTF-group J

RTFG factors throughJG, hence through

JRTFG . Thus, we obtain a surjection JRTFG J

RTFG whose composite J

RTFG

JRTFG JRTFG with the natural morphism induced by the inclusionJ J is the

identity [since JRTFG is slim [cf. Proposition 1.5, (i)], and all of these maps lie

under a fixedJ]. But this implies that the natural morphism JRTFG JRTFG is an

isomorphism. In particular, we have an isomorphism of kernels Ker(JRTF-aug

JRTFG ) Ker(JRTF-aug JRTFG ). Thus, from the point of view ofverifying the

finiteness of N, we may replace J by J [and H by an appropriate smaller opensubgroup contained in J and satisfying the hypotheses of the H of (iii)]. In

particular, since JRTF-aug JRTF-aug, we may assume without loss of generalitythat the rank rJof the pro-J-solvable free group J

RTF-free [for some subset JPrimes of cardinality 2] is either 0 or > 1p(J

RTF-aug). In particular, if l J,

then either rJ= 0 or rJ=1l(J

RTF-free)> 1p(JRTF-aug) sJ.


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Now wecompute: Since is ofcardinality 2, letl be a prime =p. Then:

1l(JRTF-free) =1l(J

RTF-free) + 1l(JRTF-aug) 1l(J

RTFG )

=1l(JRTF) 1l(JRTFG ) =1l(J) 1l(JG)

=1p(J) 1p(JG) =

1p(J

RTF) 1p(JRTFG )

=1p(JRTF-free) + 1p(J

RTF-aug) 1p(JRTFG ) =

1p(J

RTF-free) + sJ

where we apply the independence oflof Theorem 2.6, (ii). Thus, we concludethat sJ =

1l(J

RTF-free) 1p(JRTF-free) where 1l(J

RTF-free), 1p(JRTF-free)

{0, rJ}[depending on whether or not l,p belong to J] is apositive integer. Butthis implies that 0< sJ {0, rJ, rJ}, hence that sJ=rJ>0 in contradictionto the inequality sJ < rJ [which holds if rJ > 0]. This completes the proof of

assertion (iv).

Remark 2.11.1. One way of thinking about the content of Theorem 2.11,(iv), is that it asserts that aug-free decompositions of maximal pro-RTF-quotientsplay an analogous [though somewhat more complicated] role for absoluteGalois groups of MLFs to the role played bytorsion-free abelianizations forabsolute Galois groups ofFFs [cf. Theorem 2.6, (i)].

Corollary 2.12. (Group-theoretic Semi-absoluteness via Maximal Pro-RTF-quotients) For i = 1, 2, let 1 i i Gi 1, ki, Xi, i, :1 2 be as in Proposition 2.5. Suppose further that ki is an MLF; Xi is ahyperbolic orbicurve; i=. Also, fori= 1, 2, let us write

i i

for themaximal almost pro-prime topologically finitely generated closed normalsubgroup ofi, whenever a unique such maximal subgroup exists; if there does not

exist a unique such maximal subgroup, then we set idef= {1}. Suppose that is

absolute. Then:

(i) For i = 1, 2, i i; i = {1} if and only if i is of cardinalityone; ifi = {1}, theni = i. Finally, (1) 2 [so induces a morphism1/1 2/2].

(ii) In the notation of Theorem 2.11, is semi-absolute [or, equivalently,pre-semi-absolute cf. Proposition 2.5, (iii)] if and only if the following [group-theoretic] condition holds:

(s-ab) For i = 1, 2, let Hi i/i be an open subgroup such that HRTFi is

of aug-free type, and [the morphism induced by] maps H1 into H2.Then the open homomorphism

HRTF1 HRTF2

induced by mapsHRTF-free1 into HRTF-free2 .


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(iii) If, moreover, 2 is ofcardinality 2, then issemi-absolute if andonly if it isstrictly semi-absolute[or, equivalently,pre-strictly semi-absolute cf. Proposition 2.5, (v)].

Proof. First, we consider assertion (i). By Theorem 2.6, (iv), any almost pro-prime topologically finitely generated closed normal subgroup of i hence, inparticular, i is contained in i. Thus, by Theorem 2.11, (i), (ii), i = {1}if and only if i is of cardinality one; if i = {1}, then i = i. Now to showthat (1) 2, it suffices to consider the case where (1) = {1} [so 1 is ofcardinality one]. Since is absolute, it follows that (1) is normal in some opensubgroup of 2. Thus, by Theorem 2.6, (iv), we have (1) 2, so we mayassume that 2 ={1} [which implies that 2 is of cardinality 2]. But then theelasticityof 2 [cf. Proposition 2.3, (i)] implies that (1) is an open subgroup

of 2, hence that (1) is almost pro-2 [for some 2 of cardinality 2], whichcontradicts the fact that (1) is almost pro-1 [for some 1 ofcardinality one].This completes the proof of assertion (i).

Next, we consider assertion (ii). By Proposition 2.5, (iii), one may replacethe term semi-absolute in assertion (ii) by the term pre-semi-absolute. Byassertion (i), for i = 1, 2, either i ={1}or i = i; in either case, it follows fromTheorem 2.11, (iv) [cf. also Proposition 1.5, (i), (ii)], that [in the notation of (s-ab)]

the projection HRTFi HRTF-augi may be identified with the projection H

RTFi

(Hi)RTFGi

[which is an isomorphismwhenever i = i]. Thus, the condition (s-ab)

may be thought of as the condition that the morphismHRTF1 HRTF2 becompatible

with the projection morphisms HRTFi (Hi)RTFGi . From this point of view, itfollows immediately that thesemi-absolutenessof implies (s-ab), and that (s-ab)implies [in light of the existence ofH1, H2 cf. Theorem 2.11, (ii)] the pre-semi-absolutenessof. Assertion (iii) follows from Proposition 2.5, (iv), (v).

Remark 2.12.1. The criterion of Corollary 2.12, (ii), may be thought of as agroup-theoreticHom-version, in the case ofhyperbolic orbicurves, of thenumericalcriterion(H)/() = [ :H] of Theorem 2.6, (v). Alternatively [cf. the pointof view of Remark 2.11.1], this criterion of Corollary 2.12, (ii), may be thought of

as a [necessarily cf. Example 2.13 below!] somewhat more complicatedversionfor MLFsof the latter portion of Corollary 2.8, (i), in the case ofFFsor NFs.

Example 2.13. A Non-pre-semi-absolute Absolute Homomorphism.

(i) In the situation of Theorem 2.11, suppose that = Primes. Fix a naturalnumberN[which one wants to think of as beinglarge]. By replacing by an opensubgroup of , we may assume that satisfies the hypotheses of the subgroup Hof Theorem 2.11, (iii), and that the dual graph of the special fiber ofX isnot a tree[cf. the discussion preceding [Mzk6], Lemma 2.4]. Thus, we have a

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Topics in Absolute Anabelian Geometry I