Automorphism Groups of Trees Acting Locally with Affine Permutations

Automorphism Groups of Trees Acting Locally withAfne Permutations

NADIA BENAKLI1 and YAIR GLASNER21Department of Mathematics, Columbia University, New York, NY 10027, U.S.A.e-mail: [email protected] of Mathematics, The Hebrew University, Jerusalem 91904, Israel.e-mail: [email protected]

(Received: 26 February 2000)

Abstract. We consider the structure of groups that act on a pn-regular tree in a vertex transitiveway with the local action (i.e. the action of the vertex stabilizer on the link) isomorphic tothe group of afne transformations on a nite afne line.

Mathematics Subject Classications (2000). Primary 20E; Secondary 05C25, 22E40, 20F32

Key words: tree automorphisms, lattices.

1. Introduction

Let T be the q-regular tree, and H a closed subgroup of automorphisms of T actingtransitively on the vertices T . The stabilizer of a vertex acts on the link of the vertexvia a nite permutation group say G < Symq which is independent of the givenvertex. We say that the local action of H is given by the nite permutation groupG. Given a nite permutation group G, Burger and Mozes ([BMa]) dene a universalgroup UG < AutT containing a conjugate of every vertex transitive group Hwhose local action is given by G.Of special interest are the groups UG where G is a two-transitive permutation

group. In fact, for every vertex x the stabilizer UGx acts transitively on theboundary @T iff the group G is two-transitive.Let G be a two-transitive permutation group and let G0 be the stabilizer of a point.

Investigating the subgroup structure of UG, Burger and Mozes show that if thegroup G0 is non-Abelian simple (or close to being non-Abelian simple) then thefamily of closed vertex transitive groups whose local action is described by G is veryrestricted:

THEOREM 1.1 (Burger-Mozes [BMa]). Let G < Symq be a two transitivepermutation group, with G0 non-Abelian simple. Then there is a nite number of closedvertex transitive subgroups of UG whose local action is given by G. All such propersubgroups are discrete.

Geometriae Dedicata 89: 1^24, 2002. 1# 2002 Kluwer Academic Publishers. Printed in the Netherlands.

On the other hand, if G0 is solvable there can be many closed vertex transitivegroups whose action is locally described by G. An example for this is the groupG PGL2Fp acting on the projective line P1Fp. Indeed, if we take K to be anycomplete discretely valued eld with residue eld Fp, then the action ofPGL2K on its Bruhat^Tits tree is locally described by G.In this paper we attempt to investigate the extreme situation where the group G0 is

very far from being non-Abelian simple. We consider the groupA AffFq of afnetransformations acting on the afne line of some nite eld. We use the facts thatA isstrictly two transitive permutation group and that A0 is Abelian.*The paper is organized as follows. Section 2 gives some denitions and

describes some technical tools. The most important idea introduced in this sec-tion is the notion of the local permutation map associated with anautomorphism and its connection with the commensurability group of the tree.This idea was introduced by Lubotzky^Mozes^Zimmer in ([LMZ94) and isreproduced here for the convenience of the reader. In Section 4 the groupUA is introduced and some basic properties are mentioned. In Section 4the main technical tool of this paper (Theorem 4.4) is developed. This theoremenables us to combinatorially analyze the orbits for the action of cyclicsubgroups of the vertex stabilizer UAx0 on the tree. In Sections 5 and 6we draw some corollaries of Theorem 4.4 among them:

. The group UAx0 is a pro-solvable group. We give an explicit description of itspro-Sylow subgroups.

. We give an explicit description of the commensurator of a discrete subgroup ofPGL2Qp which is not a lattice.

. We give an example of an interesting closed vertex transitive subgroup of UAwhose local action is described by A. This is the junction group dened inDenition 3.6. We prove that this group has virtually torsion-free vertexstabilizers.

. Irreducible uniform lattices in AutT1 T2 whose action on both trees islocally described by primitive permutation groups are extensively studiedby Burger, Mozes and Zimmer in ([BMb, BMa, BM97, BMZ]). In Section 6we give a new constructions of such lattices using the combinatorial methodsdeveloped earlier.

2. Local Permutation Maps and Commensurator Elements

The main technical tool used in this paper is the description of an automorphism of atree using the local permutation that it induces on every vertex link. This descriptionof elements in AutT was introduced in ([LMZ94]) where it is also shown that thisdescription of automorphisms is especially suitable for describing elements of

*In fact these two properties characterize the groups AffFq.

2 NADIA BENAKLI AND YAIR GLASNER

the commensurability subgroup as these give rise to periodic local permutationmaps. In this section we recall the denitions and main results concerning localpermutation maps. We refer the reader to ([LMZ94]) for more details on the subject.We will use the following terminology

DEFINITION 2.1. A graph X is a 5-tuple X VX ;EX ; i; t; with VX the setof vertices, EX the set of (oriented) edges, i; t:EX ! VX the initial and terminalmaps, :EX ! EX a xed point free involution called the edge inversion takinge! e and satisfying ie te and te ie. A path in a graph is a sequence ofedges a e0; e1; . . . ; eN with tei iei1 8i 0; . . . ;N 1, we denoteia ie0 and ta teN . We say that a has a backtracking if for some iwe have ei ei1. A path a is called a loop if ia ta. A graph X is calleda tree, if every closed loop without backtracking is trivial (i.e. has no edges).We dene the star of a vertex Stv t1v for v 2 VX . A graph morphisml:X ! Y consists of two maps l:VX ! VY and l:EX ! EY consistent withthe structure maps i; t;.

DEFINITION 2.2. A legal edge coloring of a regular graph X is an edge coloringsuch that every edge has the same color as its opposite, and for every vertexv 2 VX the coloring induces a bijection between the edges in Stv and the set ofcolors.

DEFINITION 2.3. A V-structure on a graph X is a map f:VX ! G into a nitegroup G.

DEFINITION 2.4. A C-structure on a graph X is a map c:EX ! G into a nitegroup G, satisfying the reversibility condition ce ce1.

DEFINITION 2.5. Let X ; x0 be a q-regular graph with a base point, F a set ofcolors of cardinality q. Then a legal V-structure on X ; x0 is dened to be a tripletX ; h;f with h:EX ! F a legal edge coloring and with f:VX ! SymF aV -structure satisfying fiehe ftehe. Similarly a legal C-structure is a tripletX ; h;c with c:EX ! SymF a F -structure satisfying the consistency conditioncehe he.

DEFINITION 2.6. If l:X ! Y is a graph morphism and f:VY ! G is aV -structure (resp. c:EY ! G is a C-structure) on Y . Then we dene a V -structure(resp. C-structure) ifdef f i:EX ! G on X called the pull back of theV-structure (resp. C-structure) to X.

Notice that for covering morphisms of graphs the pull back of a legal V -structure(resp. C-structure) is still legal with respect to the pull back of the coloring.

AFFINE AUTOMORPHISMS OF TREES 3

DEFINITION 2.7. If f is a given V -structure on a graph X then the differential of fis the C-structure on the same graph dened by

dfe def fie1 fte: 2:1

Notice that the differential of a legal V -structure is a legal C-structure.

DEFINITION 2.8. Let X ; x0 be a pointed graph and c:EX ! G aC-structure. Wedene a homomorphism called the homomorphism induced on the fundamental group~cc: p1X ; x0 ! G dened by e1; e2; . . . eN ! ce1ce2 . . .ceN .

DEFINITION 2.9. We say that a C-structure c:EX ; x0 ! G on a pointed graphsatises the Kirchoff loop law if the homomorphism ~cc: p1X ; x0 ! G is trivial.

PROPOSITION 2.10. A C-structure c on a nite pointed connected graph Xsatises the Kirchoff loop law iff it is obtained as a differential of some V -structuref on the same graph.Proof.Assume that c satises the Kirchoff loop law. We dene a V -structure f in

the following way. Choose arbitrarily fx0 2 G and for every x 2 VX denefxdef ce1ce2 . . .ceN , where e1; e2; . . . eN is a path from x0 to x. The fact that csatises the Kirchoff loop law shows that this does not depend on the choice ofthe path. It is now easy to check that indeed c df. The other direction is eveneasier. &

The following lemma gives the connection between the above denitions, its proofis immediate.

LEMMA 2.11. Given a covering morphism of two pointed graphs p: X ; x0 ! Y ; y0and given a C-structure c:EY ; y0 ! G on the graph Y then the following diagram iscommutative.

Lemma (2.11) gives us the following corollary:

COROLLARY 2.12.Given a C-structurec:EY ; y0 ! Gon a graph Y. There existsa nite regular cover p: X ; x0 ! Y ; y0 such that the induced C-structure pc on Xsatises the Kirchoff loop law


Proof. Let L ker ~cc / p1Y ; y0 and X ; x0 the covering of Y ; y0 correspond-ing toL (via the Galois correspondence between subgroups of p1Y ; y0 and coveringspaces of Y ). Let p: X ; x0 ! Y ; y0 be the covering morphism. The fact that pcsatises the Kirchoff loop law now follows from the denition of L and fromLemma 2.11. The cover is regular because L is a normal subgroup. &

Let T T q be the q-regular tree. We choose a base vertex t0 2 VT . We let F be anite set of cardinality q of which we think as a set of colors. And we choose, onceand for all, a legal edge coloring h:ET ! F of T .

DEFINITION 2.13 ([LMZ94]). Given an automorphism a 2 AutT , there is a legalV -structure T ; h;fha called the local permutation map induced by a, where fha is amap

fha : VT ! SymF v! fha;v 2:2

dened by the equation:

fha;v h a hjStv1: 2:3

If the coloring h in the above denition is understood we will omit it from thenotation.

The point is that once we know the image of one vertex, the automorphisma 2 AutT is completely determined by its local permutation map. So every elementa 2 AutT can be reconstructed from the following data:(1) The initial data: The image of the base point at0.(2) The local data: The local permutation map fa.

The element a can be reconstructed from the differential of fa as well, but with somemore initial data. Namely we need:

(1) The image of the base point at0.(2) The local action on the star of the base point fa;t0.(3) The dierential of the local permutation map dfa.

One of the fundamental observations in [LMZ94] is that elements in the commen-surability group of T can be characterized as those admitting a periodic localpermutation map. This makes the combinatorial description of commensuratorelements possible.Let G < AutT be the lattice of color preserving automorphisms of T and

C CommAutT G CommT its commensurator in the whole automorphismgroup. By Leightons theorem ([Lei82, BK90]) C depends on G only up to con-jugation. Note that if L < G acts on T without inversion (i.e. L is torsion free) then


the legal edge coloring h:ET ! F factors through the covering morphismT ! LnT .

THEOREM 2.14 ([LMZ94]).An element c 2 AutT is in the commensurator C if andonly if the function fc is periodic, i.e. iff there exists a uniform torsion free latticeL < G such that the map fc:VT ! SymF factors through the covering mapT ! LnT. This holds iff the differential dfc is also periodic.Proof. The only part not proven in ([LMZ94]) is the equivalence of the third

assertion. If fc is periodic then dfc must also be periodic (with respect to the samelattice L) since the differential commutes with the projection map. Conversely ifdfc factors through T ! LnT , Corollary 2.12 gives us that a nite index subgroupL0 < L such that dfc satises the Kirchoff loop law on L

0 n T . Proposition 2.10now assures us that dfc is the differential of some V -structure on L

0 n T , the latermust coincide with fc. &

Given an element c 2 C we choose a lattice L as in Theorem 2.14 and weobtain a pointed nite graph y; y0 def L n T ;Lt0. The edge coloring h is pre-served by L so it induces a legal edge coloring of Y , also denoted by h. By ourchoice of L, fc induces a legal V -structure on Y , which will again be denotedby fc. Thus we have associated to every element c 2 C a nite graph witha legal V -structure Y ; h;fc. Note that the graph Y is not determined uniquelyby c, as we can always take a subgroup of L thereby passing to a nite coverX ; x0 of Y ; y0. The V -structure and C-structure induced on X by c willbe the same as the pull back of the corresponding structures on Y . So thetwo structures X ; h;fc and Y ; h;fc are equivalent in the sense of thefollowing denition.

DEFINITION 2.15. Two pointed graphs X ; x0 and Y ; y0 with a given coveringmap p: X ; x0 ! Y ; y0 and with two given legal edge colorings hX ; hY andV -structures fX ;fY are said to be equivalent if p

hY hX and pfY fX .

If p: T ; t0 ! Y ; y0 is the universal covering we can retrieve the localpermutation map on T induced by c, by pulling back its projection to Y , thusreconstructing the original legal V -structure T ; h;fc and (up to the ambiguityin choosing the image of the base point) the original automorphism c of the tree.The above discussion can be summarized in the following theorem, asserting an

equivalence of three sets of data.

THEOREM 2.16 The following three sets of data are equivalent.

(1) An element c of the commensurator C.(2) A legal V-structure on a nite pointed graph, together with a choice ofa base point t0

and its image by c, ct0, on the universal cover of the graph.


(3) A legal C-structure on a nite pointed graph, together with a base point t0, its imagect0 on the universal cover of the graph, and a permutation fc;t0 2 SymF .

Remark: It is easy to see that if for an element a 2 AutT we havefa;VT < G < SymF then the image of the differential of the local permutationmap is also contained in G. The converse is true if we assume that fa;v 2 G forat least one v 2 VT .

3. The Group UA of Locally Ane AutomorphismsThe main object of interest in this paper is the group UA dened below. In thischapter we introduce this group and some of its basic properties.

DEFINITION 3.1 ([BMb, BMa]) IfG < SymF is a permutation group on the set ofcolors then the universal group associated with G is dened by

UG def fa 2 AutT j Imfha < Gg: 3:1

We also dene

CG def UG \ C: 3:2

Let q pn be some xed prime power, and F Fq be the corresponding nite eld.

DEFINITION 3.2. The afne group of the eld A AffF is the group of all per-mutations on the elements of F induced by afne transformations.

A fx! ax b j a 2 F; b 2 Fg: 3:3

We denote by P and M the subgroups of A isomorphic to the additive andmultiplicative groups of F respectively. So P hPy 2 A j Pyx x yiy2F andM hMy 2 A jMyx xyiy2F . Notice that A is isomorphic to the semi-directproduct M j P, with the action Ma Pb Ma1 Pab. We let w:A!M be theprojection map. It is well known that the action of A on F is strictly two-transitive.The stabilizer of a point for the action of A is a conjugate of M and is thereforeAbelian.From now on we assume that T T q is the q-regular tree. We choose q colors

corresponding to the elements of F and a legal coloring h:ET ! F. As beforewe let G < AutT be the lattice of color preserving automorphisms of T , andlet C CommAutT G CommT . CA < C is the group of all commensuratorelements that induce on every vertex link an afne permutation.


We choose an edge e0 of the tree (say the edge in the star of t0 which is colored 0)there is an involution s 2 G which inverts this edge. We can now write

UA sh i jUA0; 3:4

where UA0 is an index two subgroup that acts without inversion.The group UA0acts locally transitively and therefore decomposes as an amalgam

UA0 UAie0 UA0e0 UAte0: 3:5

UA0 is a pro-solvable group.From Section 2 we can describe any element of UA by its local permutation map

or by the differential of its local permutation map together with some initial data.This gives rise to the following

DEFINITION 3.3. An afne structure on a q-regular graph Y is a legal C-structureY ; h;c, with h taking values in the eld F and c taking values in A < SymF.

If Y ; h;c is such an afne structure, we can associate a new C-structure to ittaking values in the groupM, namely Y ; h; w c. If a:M ! F is the isomorphismof M with F then we denote cdef a w c so that Y ; h;c is a C-structure of Yinto the multiplicative group of F. Note that there is no loss of information by pass-ing to c, in fact, using the assumption ce 2 StabAhe, we can reconstruct c fromc. This is summarized in the following proposition.

PROPOSITION 3.4. Let a 2 A be a permutation such that a0 he, then thefollowing equation holds

ce a w ce a1: 3:6

Proof. First notice that the right-hand side of Equation (3.6) does not depend onthe choice of a as different choices of a correspond to multiplying a on the rightby elements of A0 M and the group M is Abelian. Thus, the proposition canbe proved by substituting a concrete transformation for a. If we choose a Pheand we let a1 ce a My we obtain

ce Phe My P1he Phe My P1he M1y My Pheyhe My 3:7

Which shows that wce My and completes the proof. &

Note.

. If we write cex c1x c0 then ce c1.


. Proposition 3.4 and the remarks in the previous section imply that an elementa 2 UA can be reconstructed from dfa together with some initial data ^the image of the star of a vertex.

DEFINITION 3.5. We say that an afne structure Y ; h;c satises the Kirchoffjunction law if for every vertex v 2 VY we haveY

e2Stvce 1 3:8

It is obvious that if an afne structure satises the junction law, then so does theinduced structure on every covering graph.

DEFINITION 3.6. We dene

J fc 2 UA j c satisfies the junction lawg 3:9

and we call J the junction group of UA.

It is still left to show that this is indeed a group. We leave the proof of the followingproposition to the reader.

PROPOSITION 3.7 The junction group dened above is a closed nondiscrete groupwith G < J. J acts vertex transitively on the tree and its local action is given by A.

4. The Orbit Structure of Elements in UAIn this section we give the main result of this paper. We use two properties special tothe group A ^ it acts strictly two-transitively with an Abelian stabilizer. These enableus to give a nice description for the orbits of the action of cyclic subgroups on the setof edges of the tree ^ ET .We x an element c 2 UAt0 and we wish to study the orbits for the action of hci on

the edges of the tree ET . We denote the spheres around t0 by

Sn def fv 2 VT j dv; t0 ngand the balls by

Bn def fx 2 T j dx; t0W ng:We also denote by

Rm; n def Bnn Bm fx 2 T j mW dx; t0W ng

for every n > mX 0. The orbits for the action of c on ET lie within the sets of theform Rn; n 1.


DEFINITION 4.1. Let H < StabAutT t0 be a group, O f0; f1; . . . ; fN1

ERn 1; n an orbit for the action of H on ET . Then we say that the orbitO splits into M pieces at the nth sphere if there are exactly M orbits for the actionofH on ERn; n 1 that intersect withO non trivially (on Sn). We call these orbitsthe direct descendents of O. The descendents of O are dened as the orbits ofH whichare obtained fromO by an iterated passage to direct descendents. The Shadow of O inRm 1;m for some m > n is the union of all orbits ofH in ERm 1;m which aredescendents of O.

LEMMA 4.2. Let H;O be as in Denition 4.1. For every i 2 Z=NZ, letvi tfi 2 Sn, Wi def Stvi and Orbi be the set of orbits for the action ofHvi StabH vi on Wi. Let Odef O O0;O1; . . .OMf g be a set containing O andall of its direct descendents. Then for every i 2 Z=NZ there is a bijectivecorrespondence:

O! OrbiOj ! Oj \Wi 4:1

preserving the ratio of the sizes of the orbitsProof. SinceH acts transitively on O, the action ofHvi onWi and the action ofHvj

on Wj are conjugate.Thus we restrict ourselves to the study of the action of Hv0 onW0.It is obvious that the orbits of Hv0 are subsets of the orbits of H. In the otherdirection assume that d1; d2 2W0 are two edges which are in the same H orbit,so that d2 hd1. The fact that both di are in W0 implies that h 2 Hv0 . The remarkfrom the beginning of the proof now shows that the size of Oj \Wi does not dependon i which nishes the proof of the lemma. &

For every nX 2, we dene a map zn:ERn 1; n ! EBn 1 to be the mapassociating to each edge f the unique edge znf such that:. hf hznf , i.e both edges are of the same color.. The distance df ; znf is minimal with respect to the previous property.

This is illustrated in Figure 1. We consider the following subgroup of AutT :

Gn def fg 2 StabAutT t0 j h gf h g znf 8f 2 ERn 1; ng fg 2 StabAutT t0 j zn gf g znf 8f 2 ERn 1; ng: 4:2

The question whether an element g 2 StabAutT t0 is in Gn depends only on itsrestriction to Bn. We will therefore, by abuse of notation, refer to Gn also as asubgroup of AutBn.


We notice that if g 2 AutBn, then g admits a unique extension toGn1 denoted ~gg.More precisely, we have

~ggf gf if f 2 EBnzn11 g zn1f if f 2 ERn; n 1:

4:3

Note that if g acts on the tree in a local afne way then so does its extension ~gg.

LEMMA 4.3. The group

Gn1A def fg 2 Gn1 j fg;VBn < Ag

acts transitively on Sn but its action splits into q 1 orbits at Sn.Proof. As A is a two-transitive permutation group, the group UA acts

transitively on Sn for every n (see [BMa]). Now since each element ofUAjAutBn can be extended to an element of Gn1A, it follows that the latergroup still acts transitively on Sn. By Lemma 4.2, it will be enough to show thatif an element g 2 Gn1A xes a vertex v 2 Sn then it xes Stv pointwise. Sincethe action of A on F is strictly two-transitive, it is enough to show that g xestwo edges of Stv. Let w be the vertex on Sn 1 adjacent to v. It is clear thatg xes two edges in Stw, namely the two edges on the geodesic from t0 to v.By denition of Gn1 we have fg;v fg;w so that g must also x two edges inStv. &

THEOREM 4.4. Let c 2 UAt0 , and nX 3. Let O ff0; f1; . . . ; fN1g 2 ET be anorbit for the action of c on ERn 1; n (assume that all edges are directed fromSn 1 to Sn). We dene an element x xO 2 F

x xO defYN1j0

dfc;fj : 4:4

Then O splits into q 1=ordx orbits of equal sizes at the nth sphere.Proof. Let vj tfj 2 Sn, and Wj Stvj. Without loss of generality we may

assume that cfj fj1 8j 2 Z=NZ.Let us associate with each one of the q-orbits for the action of Gn1 on

ERn 1; n 1 (cf. Lemma 4.3) an element of the eld F by taking

Figure 1. Denition of the mapping zn.


O! hO \W0. We now dene a new edge coloring taking each edge inERn 1; n 1 to the color associated with its orbit:

h:ERn 1; n 1 ! F 4:5e! hGn1Ae \W0 4:6

We have hjW0 hjW0 . We may assume without loss of generality that the notation issuch that hjRn1;n 0. Now we have two colorings of ERn 1; n 1 and c inducestwo corresponding local permutation maps fhc and f

hc on the vertices of Sn. By

denition, if d 2 Gn1A and if dv0 vj for some j 2 Z=NZ, then:hjWj hjW0 d1jWj hjW0 d1jWj fhd1;tfj h: 4:7

By Lemma 4.2 applied to H hci, it will be enough to show that cN acts on W0with q 1=ordx 1 orbits: one singleton and the rest of equal sizes. The actionof fhcN ;v0 on F is isomorphic to the action of c

N on W0. We decompose the formerpermutation as a product:

fhcN ;v0 fhc;vN1 fhc;vN2 fhc;v0 : 4:8

We dene the element d gcjBmcjBm to be the unique extension of cjBm to Gn1. UsingEquation (4.7) we obtain the following formula for each one of the factors inEquation (4.8)

fhc;vj h c hjWj 1

fhdj1;v0 1 h c hjWj 1 fhdj ;v0 fhdj1;v0 1 fhc;vj fhdj ;v0 fhdj ;v01fhd;vj 1 fhc;vj fhdj ;v0 fhdj ;v01dffj fhdj ;v0Mxfj 4:9

By denition the local permutation that d induces on Stvj Sttfj is identical tothe permutation that c induces on Stifj. This explains the equality before the lastone in Equation (4.9). The last equality follows from Proposition 3.4.Substituting Equation (4.9) into Equation (4.8) we get a formula for the action of

cN on W0

fhcN ;v0x fhc;vN1 fhc;vN2 fhc;v0xMxfN1 MxfN2 Mxf0Mx: 4:10


So the action is isomorphic to multiplication by x on F. Multiplication by x acts on Fwith one orbit of size one and q 1=ordxorbits of size ordx, which completes theproof.

5. Applications of the Orbit Structure Theorem

In this section we give some corollaries of the orbit structure Theorem 4.4: Wecalculate the pro-Sylow groups of UAt0 , we discuss the structure of the junctiongroup and we construct some examples of commensurator elements with largeorbits. These allow us later to construct explicit examples of irreducible latticesin products of two trees.

5.1. THE PRO-SYLOW GROUPS OF UAt0The group UAt0 is pro-solvable and thus decomposes as a product of its pro-Sylowsubgroups. Furthermore it is easy to determine exactly which primes occur bywriting UAt0 as an inverse limit. Here we wish to give an explicit descriptionof the pro-Sylow subgroups of UAt0 in terms of local permutation maps. We rstdeal with the normal subgroup K StabUAStt0 /UAt0 , the point stabilizerof Stt0.

THEOREM 5.1. Let L < F be a subgroup and dene

L^L fa 2 K jdfa;e 2 L 8e 2 ETg 5:1

(1) L^L is a closed subgroup of UAt0.(2) If l is a prime and L is an l-subgroup of F then L^L is a pro-l subgroup of K.(3) If L is an l-Sylow subgroup of F then L^L is a pro-l-Sylow subgroup of K.

Proof. It is easy to verify that the following cocycle equation holds for every edgee 2 ET :

dfab;e dfa;be dfb;e 5:2which implies that L^L is a group whenever L is. If a 62 L^L then there exists a specic edgee 2 ET such that dfa;e 62 L. This later condition is open, so L^Lmust be closed, proving(1). Assume now that L is an l-group. Since L^L is closed it is a pro-nite group, itfollows that:

L^L lim

L^LjBi 5:3

So we only have to check that L^LjBi is an l-group for every i 2N. To prove this it isenough to show that any element has an order which is a power of l. We pickan element a 2 L^L and show that when it acts on the sphere Si it acts with orbitsall of whose cardinalities are powers of l. This is true for i 1 because by assumption


all orbits there are singletons. We proceed by induction on i. We havedfa;e 2 L 8e 2 ET . The induction hypothesis is that any a orbit, O onERi 1; i consists of lr points for some r. By Theorem 4.4 any such orbit splitsinto orbits of size lr ordx on the ith sphere, where x Pf2O dfa;f . By ourassumptions dfa;f 2 L 8f 2 O which implies x 2 L, thus xmust have an order whichis a power of l. This completes the proof of (2). To prove the last assertion we expressF as a (direct) product of Sylow subgroups F L1 L2 Lm and show thatK is the product of the corresponding pro li Sylow subgroups:K L^1L1L^2L2 L^mLm. Let k 2 K be given. We have to nd ki 2 L^iLi such thatk kmkm1 k1. Recall that by Section 2 and by Proposition 3.4 it is enoughto specify the differential of the local permutation map of each one of these kisin order to dene them. We write dfk;e lk;e;mlk;e;m1 lk;e;1, where lk;e;i 2 Li.Now dene k1 to be the element satisfying dfk1;e lk;e;1 then dene recursivelyki by the equation dfki;ki1ki2k1e lk;e;i The cocycle equation (5.2) proves thatk kmkm1 k1. This completes the proof. &

Let us recall that UAt0 KA. For any subgroup B < A and any prime l we cannd a pro-l-Sylow subgroup of KB which factors as a product of a pro-l-Sylowsubgroup of K and an l-Sylow subgroup of B this can be applied to the groupUAt0 itself and also to the stabilizer of an edge UAe0 to give the followingcorollaries:

COROLLARY 5.2.. The group UAe0 admits pro-l-Sylow subgroups iff l divides jFj q 1.. The group UAv0 admits pro-l-Sylow subgroups iff l divides jAj qq 1.

Furthermore it admits an innite pro-l-Sylow subgroups iff l dividesjFj q 1.

5.2. AN APPLICATION TO PGL2LThere is a strong connection between the nite groups A AffF and PGL2F. Theformer being the stabilizer of a point for the action of the later on the projective lineover F. The group UPGL2F contains many closed, vertex transitive subgroupswhose local action is given by PGL2F. An example is G PGL2L (where L isa non-Archimedean local eld with residue eld F), acting on its Bruhat^Tits treeY . In this section we make use of the connection between A and PGL2F in orderto calculate the commensurator in G of a discrete subgroup of PGL2L which isnot a lattice.Let y0 2 VY be a base vertex. We can color in a natural way the star of y0 with a

set of colors corresponding to the projective line P1F F [ f1g in such a way thatthe action of Gy0 on this star factors through the action of PGL2F on P1F. Denotethis coloring by Sty0 feZgZ2P1F. We now wish to extend this coloring to a legalcoloring of EY in such a way that G < UPGL2F. Choose a set of involutions


gZ 2 G inverting the edges eZ. The group L hgZiZ2P1F will be a lattice acting simplytransitively on the oriented edges of Y . Thus L denes an edge coloring of Y ^coloring an edge e with the same color as the edge Le \ Sty0.Wemight hope to obtain an interesting subgroup ofUA in the following way. Let

T Y be the connected component of y0 in the forest which is obtained by deletingall the edges in Y colored by 1. T is a jFj-regular tree, it inherits from Y a legalcoloring by a set of colors corresponding to the elements of F. Let H < G bethe subgroup of G which xes T as a set. If jFj > 2 the map H ! AutT is injectivesince an element of G is determined by its action on any three points of @Y . For everyvertex v 2 VT and any element of H the local permutation map fh;v xes1, we cantherefore view H as a subgroup of UA.The groupH contains the group G hgZiZ2F which is a lattice inAutT . In fact,H

even contains the commensurator of G in G:

LEMMA 5.3 CommGG < H.Proof. CommGG is generated as a group by G and a vertex stabilizer

CommGGy0 . Since G < H, it is enough to prove that CommGGy0 < H. Letc 2 CommGGy0 , pick S < G of nite index such that cSc1 < G. We observe that:

cSy0 cSc1cy0 Gcy0 Gy0 T 5:4

from this and from the fact that the convex hull of Sy0 is T it follows that cT T .&

We now calculate H explicitly thus showing that the commensurator is small:

PROPOSITION 5.4. H : G

the notation of Theorem 4.4).

xO YM1k0

xOk: 5:5

This enables us to conclude a few facts about the junction group.

COROLLARY 5.6. Assume that c 2 CAt0 satises the junction law. Suppose that cadmits an orbit O ff0; f1; . . . ; fN1g 2 ERn; n 1 for some nX 2 with xO agenerator of the cyclic group F. Then the orbit O never splits, i.e. c acts transitivelyon the shadow of O on the sphere Sm for every mX n.Proof. We prove this by induction using Theorem 4.4 in every step. Theorem 4.4

implies that O On does not split in the nth sphere so it has only one descendentorbit On 1 ERn 1; n 2. The junction law now implies that for this orbitagain xOn 1 x. &

COROLLARY 5.7. The junction group J has virtually torsion free vertex stabilizers.More specically the subgroup of J which xes pointwise the set Stt0 is torsion free.Proof.We pick an element c 2 Jt0 which xes pointwise Stt0. (Note that for such

an element the proof of Theorem 4.4 works also in the case n 2.) It is enough toprove that either c id or c admits orbits of arbitrarily large cardinality. Ifc 6 id then there exists at least one edge e 2 ET with xe 6 1, such an edge of mini-mal distance from t0 will also be xed under the action of c. This edge now splitsinto orbits of size larger then one. The junction law implies that one of thesenew orbits again carries a x 6 1 which means that it will again split into larger orbitsand we continue by induction. &

5.4. CALCULATING THE ORBITS FOR A SPECIFIC COMMENSURATOR ELEMENT

In this section we use Theorem 4.4 to give explicit examples for commensuratorelements which act with few orbits on the boundary of the tree. This enables usin the next section to give explicit constructions for irreducible lattices acting onthe product of two trees.In a typical example we describe an element c 2 CA by giving its local

permutation map fc, the differential of the local permutation map dfc or theprojection of the later on the multiplicative group of the eld dfc, each with thenecessary initial data. Since by assumption c is an element of the commensuratorall this information can be given by V -structures or C-structures on nite graphs.We x once and for all a q-regular tree with a base point T ; t0 and a legal coloring

h:ET ! F (where F is the nite eld of q elements). We identify T with the universalcover of all our pointed nite graphs. For convenience we will work only withelements c 2 CAt0 so that we always have ct0 t0, such an element is described


by one of the following (graphic) notations (these notations are illustrated for aspecic commensurator element in Figure 2).

. A pointed graph X ; x0 with the legal coloring h:EX ! F drawn on the edges,and the local permutation map fc:VX ! A drawn on the vertices.

. A pointed graph X ; x0, with the legal edge coloring h : EX ! F and thedifferential of the local permutation map, dfc:EX ! A drawn on every edgeas a pair h; dfc. Together with fc;t0 .

. A pointed graph X ; x0, with the legal edge coloring h:EX ! F and themultiplicative part of the differential of the local permutation map,dfc a w dfc:EX ! A, drawn on every edge as a pair h; dfc. Togetherwith fc;t0 .

PROPOSITION 5.8. Let c be the element of CAt0 dened by any of the threeequivalent descriptions in Figure 2. If the following conditions hold*

(1) s xes 1 2 F and acts as a q 1-cycle on the rest of the elements of the eld.(2) The element ydef a ws1 t is a generator of the cyclic group F.Then ch i, the closure of the cyclic group generated by c, xes the unique edge ~ee in Stt0with h~ee 1, and acts transitively on the boundary of each one of the two connectedcomponents of T n ~ee.Proof. The proof uses induction. As the two halves of the tree around ~ee are

symmetric, we will prove the theorem only for the half containing t0. When usingthe notations Sm;Bm;Rm; n we will understand (for the purpose of this proofonly) the intersection of the respective sets with the relevant half tree.It is obvious that the assumption of the theorem about s implies that c xes ~ee and

acts transitively on Stt0~ee, i.e. c acts transitively on S1, thus all the edges inR0; 1 form one c-orbit denoted O1. We can easily verify that xO1 y 2 F isa generator of the multiplicative group, where xO1 is dened as inTheorem 4.4. Theorem 4.4 (which holds in this situation also for n 2) now impliesthat c acts transitively on the edges of R1; 2. We denote this orbit by O2 and con-tinue by induction. The only thing we need for the induction step is the followingcombinatorial lemma.

LEMMA 5.9. Let xndefQ

f2ERn1;n dfc;f 2 F, then xn is a generator of the cyclicgroup F.Proof.The tree T is the universal covering of the graph in Figure 3. We give a label

and a sign to each directed edge of the graph in the following way. Every edge on theleft (resp. right) side of the graph is getting the sign (resp. ). Each edge labeled 0and directed from left to right (resp. right to left) gets the sign (resp. ). Thedirected edges of T acquire labels and signs via the covering morphism. The edge

*The values specified in Figure 2 are examples for a choice of elements which satisfy theconditions of the theorem.


~ee covers the edge labeled 1 on the left and is therefore labeled 1;. We deneJn; j def fnumber of edges labeled j, directed from Sn 1 to Sn and countedwith their signs g. We obtain the following:

J1; 0 1;J1; 1 0;J1; j 1 8j 2 F n f0; 1g;Jn 1; 0 Jn; 1 q 2Jn; z;Jn 1; 1 Jn; 0 q 2Jn; zJn 1; j Jn; 0 Jn; 1 q 3Jn; z 8j 2 F n f0; 1g: 5:6

Here we used twice the fact that there is a symmetry between all labels different from0 and 1. We use z in the above equations to denote any element of the eld which isneither 0 nor 1, we could have replaced this by any other element. Using

Figure 2. The equivalent denitions for the element c.

Figure 3. Labeled oriented graph.


Equations (5.6), it is easy to prove by induction the following formula

Jn 1; 0 Jn 1; 1 Jn 1; z Jn; z Jn; 1 Jn; 0 1n modq 1: 5:7

Thus, xn yJn;0 is always congruent to y or to y1 which proves the lemma and theproposition. &

6. Examples of Lattices in Products of Two Trees

Recently, much work was done on uniform lattices in the automorphism group of theproduct of two trees. In a series of papers, Burger, Mozes and Zimmer show thatthese lattices have many properties reminiscent of lattices in higher rank Lie groups([BMb, BM97, BMZ]). As in the case of Lie groups a notion of an irreducible latticeis necessary.In the setting of semisimple Lie groups there is a dichotomy: A lattice in the

product of two simple Lie groups L < G2 G2 has projections which are either dis-crete (a reducible lattice) or dense (an irreducible lattice). In products of two treesthis dichotomy no longer holds, in fact it turns out that a uniform latticeG < AutD can never have dense projections (see [BMb]). Still we would like tothink of an irreducible lattice as such a lattice which has large projections.A family of denitions was given by Burger, Mozes and Zimmer all of which require,in a stronger or weaker sense, that the projections of the lattice be big.

DEFINITION 6.1. Let T1;T2 be two regular trees, D def T1 T2 their product (atwo dimensional square complex), pri : AutD ! AutTi i2f1;2g the projections,G < AutD a uniform lattice and Hi def priG the closures of the projectionsof the lattice. We say that G is an irreducible lattice if both Hi are nondiscrete.

Remark 6.2. By a lemma of Burger, Mozes and Zimmer, G is irreducible iff any oneof the His is nondiscrete (see [BMb]).

DEFINITION 6.3. A closed subgroupH < AutT of the automorphism group of atree is called locally primitive, if for every vertex x of T , StabHx acts as a primitivepermutation group on the link Lkx.

DEFINITION 6.4. A closed groupH < AutT in the automorphism group of a treeis called locally innitely transitive, if for every vertex x of T , StabH x acts as atransitive permutation group on the sphere Sx;m for every m 2N.

Stronger notions of irreducibility for G are obtained by requiring the projectionclosures Hi to be non discrete and locally primitive, or locally innitely transitive.


Many theorems have been proved ([BMb]) in this setting showing many similarities,but also many differences, between the theory of irreducible lattices in the productof two regular trees (in each of the above settings) and the theory of irreduciblelattices in semisimple Lie groups.In this paper we will give a method for construction of such lattices using the

techniques discussed in the earlier sections.It turns out that there is a strong connection between the commensurator of a

lattice G < AutT1 and embedding of G into lattices in products of trees. As anindication we bring the following facts.

PROPOSITION 6.5. LetG < AutD be a lattice in the product of two trees (using thenotations of denition (6)). Considering only the action of G on T2 we obtain ^ as aquotient ^ a nite graph of groups G n nT2 where all the vertex and edge groupsare commensurable, uniform lattices in AutT1.

THEOREM 6.6. Let GX ; x0 be a nite graph of groups with all the groupsfGsgs2VX being commensurable uniform lattices in AutT1 where T1 is some uniformtree. Let Ldef p1GX ; x0 and let T2;def gGX; x0GX; x0, the Bass^Serre treecorresponding to this graph of groups. Let Ddef T1 T2. Then

(1) There is a natural action of L on D making L into a uniform lattice in AutD.(2) pr1L Gsjs 2 VXh i < AutT1 and thus L is irreducible i

Gs j s 2 VXh i < AutT1 is nondiscrete.(3) pr2L < AutT2 is given by the fundamental group of the eective quotient the

graph of groups GX ; x0. Thus L is reducible i the eective quotient of GX is a graph of nite groups. The lattice L acts locally transitively on T2 i X is asingle edge (i.e. L is an amalgamated product S S\S S0). In such a case L actslocally primitively on T2 i S \ S0 is a maximal subgroup in both S and S0.

Proof. By the universal property of the fundamental group of a graph of groupsthere is a well dened map L! AutT1 by which L acts on T1. The action ofL on T2 is the regular action of the fundamental group on the Bass^Serre tree.We let the action LnD be the diagonal one lx1; x2 lx1; lx2. In order to prove(1) we must show that L acts on D with discrete stabilizers and with a compactfundamental domain. If x x1; x2 is a vertex of D thenStabLx StabStabLx2x1. This later group is nite because StabLx2 acts onT1 as a discrete group (a uniform lattice which is conjugate to one of thefGsgs2VX ). To show that L is cocompact let Y def fysgs2VX T2 be a sectionVX ! T2 with Lys Gs. Applying some element of L to x x1; x2we may assumethat x2 yt 2 Y . Now applying an element of Gt we can move x1 into some compactfundamental domain for the action of Gt on T1, without affecting the secondcoordinate. Thus we have proved (1).Assertion (2) follows from Remark 6.2.


As for (3), the equivalence of L being reducible with GX having an effectivequotient which is a graph of nite groups again follows from Remark 6.2 Theassertion about local transitivity is obvious. As for the local primitivity we wishto check the action of StabLx2 on Lkx2 for an arbitrary vertex x2 2 VT2. By con-jugation though we may assume that x2 is the vertex stabilized by S (or S0) andthat the stabilizer of one of the edges in the link is S \ S0. The elements actingtrivially on the link are exactly those in the (normal) subgroup stabilizingtheone-sphere around x2. Dividing by this group we are reduced to the well-knownfact that a group action is primitive iff it is transitive and the point stabilizersare maximal subgroups. &

It follows from Theorem 6.6 that we can generate irreducible lattices by ndingtwo commensurable uniform lattices in S;S0 < AutT1which generate a large groupof AutT1. In this section we give an example of how this can be achieved, using themethods developed in the previous sections. Namely, we construct a family ofirreducible lattices in AutT1 T2, where T1 and T2 are two regular trees. Theselattices will act locally primitively on one of the trees and locally innitelytransitively on the other.The strategy is as follows, using Theorem 6.6 we can construct an element c 2 C of

the commensurator such that ch i acts on the boundary of T1 with a nite number oforbits. We decompose c as a product of two commensurator elements of nite orderc ba. A periodic element of the commensurator is also contained in a lattice(if it commensurates G it also normalizes a subgroup of nite index) thus we con-struct two commensurable uniform lattices Ga;Gb containing a; b respectively.By Theorem 6.6 we have an action of Ldef Ga Ga\Gb Gbon a product of two trees.The projection pr1L hGa;Gbi contains both c and a uniform lattice and it is easyto verify that it must act locally innitely transitively on T2.In ([LMZ94]) the following criterion is proven for an element of the

commensurator to be periodic.

THEOREM 6.7 ([LMZ94]). Let c 2 C be a commensurator element on the regulartree T. Then c is of nite order iff its local permutation map factors through a coveringof some nite graph T ! X in such that the two colored graphs: X ; f ! hf andX ; f ! fa;tf hf X ; f ! fa;if hf are isomorphic.

As a specic example we start with the commensurator element constructed inSection 5.4. We take l q 1 2r 1 to be a Mersenne prime,F Fq h0; 1; z; z2; . . . ; zl1i the corresponding nite eld, T1 the qregular tree,h:T1! Fq a legal coloring, G < AutT1 the lattice of color preservingautomorphisms, G0 < G the subgroup (of index 2) acting without inversions. Wecan think of G0 as the fundamental group p1X ; x0 of the graph X in Figure 4and we take L < G0 to be the index two subgroup L p1R; r0. As the cyclic groupof the eld is of prime order (an l-group) the generator z of the cyclic group has


a square root ^zp

. We consider the element c 2 C constructed in Section 5.4 which isdened by Figure 2. Figure 5 gives a decomposition of c as a product of two periodicelements c b a. The element that b describes an automorphism of the graph R oforder l. As follows from [LMZ94] this implies that b is periodic of order l. Asfor the element a, it does not describe an automorphism of the graph R but acan be described as an automorphism of the graph X as illustrated in Figure 6. Herethe action of a on the edges of X is just like the action of the permutation s d. Ashort calculation (using the fact that every element of A is of order 2 or l) showsthat a is an l-cycle for every l > 3 and that in the case l 3 it is an element of order 2.Let us notice that hG; ai < NAutT1G and that hG; bi < NAutT1L. Therefore

both these groups are lattices in AutT1. Let Ldef hG; ai G G; bh i, and let T2be the corresponding Bass^Serre tree. Let Ddef T1 T2.

PROPOSITION 6.8. With the above notations, the following hold:

(1) T2 is an l-regular tree for l > 3, and it is the barycentric subdivision of the 3-regulartree if l 3.

(2) L is an irreducible lattice in AutD.(3) L acts locally innitely transitively on T1 and locally primitively on T2.

Figure 4. The labeling of T.


Proof. In order to prove (1) we must show that: hG; aiG hG; biG l for l > 3,and that hG; aiG 2 in the case of l 3. Since hG; ai normalizes G, it maps intothe automorphism group of the graph X . As G maps into AutX trivially andthe image of hG; ai is a cyclic group of order l (resp. 2 if l 3). We get the desiredindices by pulling back to hG; ai.In a similar way, the group hG; bimaps into the automorphism group of the graph

R. Explicitly, this group is AutR Sl Sl j Z=2Z (the action here is byswitching the two components). The image of G is the factor of order two, andthe image of hbi is the group h zp 1; zp i 2 Sl Sl, which is a cyclic groupof order l, normalized by the image of G. Pulling back to hG; bi we gethG; bi : G 2l : 2 l.For the proof of (3). By Theorem 6.6 the action of L on T1 is the action of hG; a; bi.

Using the fact (Proposition 5.8) that c b a acts innitely transitively on halfspheres, and that G acts on the vertices of T1 with two orbits we get the desiredconclusion.As L acts locally transitively on T2, and since T2 is regular of prime valence, the

action of L must be locally primitive.

Figure 5. Denition of some commensurator elements.

Figure 6. Description of a as a re-coloring of X.


For the proof of (2) pr1L < AutT1 is nondiscrete and hence the lattice isirreducible by Remark 6.2. &

Acknowledgements

Both authors wish to express their thanks to Professor Hyman Bass for manydiscussions on these subjects. We also wish to thank Professor Shahar Mozesfor many helpful remarks. This paper was written while the second author wasvisiting Columbia University. He would like to thank the university, and especiallyProfessor Hyman Bass, for their warm hospitality.

References

[BK90] Bass, H. and Kulkarni, R.: Uniform tree lattices. J. Amer. Math. Soc. 3(4) (1990),843^902.

[BMa] Burger, M. and Mozes, S.: Groups acting on trees: From local to global structure.Publ. Math. IHES 92 (2000), 113^150 (2001).

[BMb] Burger, M. and Mozes, S.: Lattices in product of trees. Publ. Math. IHES 92(2000), 151^194 (2001).

[BM97] Burger, M. and Mozes, S.: Finitely presented simple groups and products of trees.C.R. Acad. Sci. Paris Se r. I. Math. 324(7) (1997), 747^752.

[BMZ] Burger, M., Mozes, S. and Zimmer, R. J.: Linear representations and arithmeticityfor lattices in AutTn AutTm, In preparation.

[Lei82] Thomson Leighton, F.: Finite common coverings of graphs, J. Combin. TheorySer. B 33(3) (1982), 231^238.

[LMZ94] Lubotzky, A., Mozes, S. and Zimmer, R. J.: Superrigidity for the commersurabilitygroup of tree lattices. Comment. Math. Helv. 69(4) (1994), 523^548.


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Automorphism Groups of Trees Acting Locally with Affine Permutations