Perfect Semigroups and Aliens

Semigroup Forum Vol. 66 (2003) 236–272c© 2002 Springer-Verlag New York Inc.

DOI: 10.1007/s002330010133

RESEARCH ARTICLE

Perfect Semigroups and Aliens

Brigitte E. Breckner∗ and Wolfgang A. F. Ruppert

Communicated by Jimmie D. Lawson

Abstract

A topologized semigroup is called perfect if its multiplication is a perfect map(= a closed continuous mapping such that the inverse image of every point iscompact). Thus a locally compact topological semigroup is perfect if and onlyif its multiplication is closed and each of its elements is compactly divided, thatis, its divisors form a compact set. In the present paper we study compactlyand non-compactly divided elements in the contexts of general locally compactsemigroups, subsemigroups of groups, Lie semigroups and subsemigroups ofSl(2,R) .

2000 Mathematics Subject Classification: 22E15, 22E67, 22E46, 22A15,22A25.Key words and phrases: Alien elements, divisors of an element, perfectsemigroups, proper semigroups, compact order intervals, Bohr compactification,subsemigroups of Sl(2,R) , Lie semigroups, Lie semialgebras, umbrella sets.

1. Introduction

An element s of a topological semigroup S is said to be compactly divided if theset {(x, y) ∈ S×S | xy = s} is compact. If s is not compactly divided (that is,if it can be factored by elements living arbitrarily far away) then we say thatit is purely alien. Finally, if for every compact subset K of S the element s iscontained in the closure of (S\K)S ∪ S(S\K) then s is called an alien.

The above three concepts appear implicitly or under other names in theliterature (cf., e.g., [12], [11], [19]), they show up in the following contexts:

• In the harmonic analysis of semisimple Lie groups we need informationabout the compactness of intervals with respect to an order induced bya Lie subsemigroup (compactness is needed for integration). Compactlydivided elements correspond to compact order intervals.

• In the general theory of locally compact topological semigroups a naturalquestion asks for conditions under which the product AB of two closedsets A and B is closed as well. If A , B are closed and Al(S) denotes theset of all aliens in S then AB ∪Al(S) is always closed.

∗This author was supported by the Deutsche Forschungsgemeinschaft.

Breckner and Ruppert 237

• Tubular neighborhoods: In the Theory of Lie semigroups one shows that ifin a Lie semigroup the group of units is compact then it has a neighborhoodbasis consisting of complements of closed ideals. As we shall see in section 4below this result can be shown much more generally for the divisor setMD(s) = {x ∈ S | s ∈ {x} ∪ xS ∪ Sx ∪ SxS} of any non-alien elementin a closed noncompact subsemigroup of a locally compact group, bystrictly ‘topological’ arguments (not even one-parameter subsemigroupsare involved).

• The Alexandrov compactification of a locally compact semigroup S ,with the element at infinity acting as a zero element, is a topologicalsemigroup if and only if S has no alien elements. Thus it is not surprisingthat the set of aliens plays an important role for the construction ofcompactifications.

In the first five sections of the present paper we try to clarify the aboveconcepts and their mutual relationship from a systematic point of view. Theaim is to establish all important facts by purely algebraico-topological methods,avoiding as much as possible any analytical machinery and special features ofthe main objects for applications, the Lie semigroups. (We hope to prove thatthis general approach is interesting by itself.)

In the remaining sections we explicitly describe the set of aliens in aninteresting special case, namely the case of subsemigroups of Sl(2,R).

2. Perfectness conditions

2.1. General assumption

Throughout this paper all topological spaces are assumed to be Hausdorff, ifnot explicitly stated otherwise.

2.2. Perfect maps

([6], p. 238ff) Recall that a continuous map f : X → Y between topologicalspaces is called perfect, if it is a closed mapping and the fiber f−1(y) of eachpoint y ∈ Y is compact.

2.3. Basic properties of perfect maps ([6])

(i) If f and g are perfect maps then so is their composition f◦g .

(ii) The inverse image of a compact subset under a perfect map is compact.

(iii) If Y is a k -space then a continuous map f : X → Y is automaticallyclosed if the inverse f−1(K) of any compact subset K of Y is compact,so in this situation the perfect maps are exactly the proper maps in thesense of Bourbaki ([2], p. 97, cf. also p. 104). However in general thecompactness of f−1(y) for all y ∈ Y does not imply that f is closed (evenif we assume X and Y to be locally compact).

238 Breckner and Ruppert

2.4. Perfectness conditions for semigroups

Let S be a topologized semigroup and write µ for the multiplication mapS × S → S . Then S is called a

(i) perfect semigroup if µ is a perfect map;

(ii) c-perfect semigroup if for every compact subset K of S the restriction ofµ to the set (K × S) ∪ (S ×K) is a perfect map;

(iii) separately perfect semigroup if, for every s ∈ S , the left and the righttranslations x �→ sx and x �→ xs are perfect maps.

We shall see below that every closed subsemigroup of a topological group isboth c-perfect and separately perfect. Thus if the reader is only interestedin closed subsemigroups of groups then he may safely replace the conditions‘c-perfect’ or ‘separately perfect’ by the condition ‘closed subsemigroup of atopological group.’ Our aim in introducing these terms is primarily to conveyan understanding of what are the properties of groups essential for our argu-ments.

2.5. Remark. (i) With the above definitions the following implications holdfor a semigroup on a topological space:

perfect =⇒ c-perfect =⇒ separately perfect

perfect, or compact c-perfect, semigroup =⇒ topological semigroup

separately perfect semigroup =⇒ semitopological semigroup.

If S is a [perfect] [c-perfect] [separately perfect] semigroup then so is everyclosed subsemigroup of S .

(ii) If S is perfect then every maximal subgroup H(e) of S is compact,since

H(e) = {x ∈ eSe | ∃y ∈ S: xy = e} ∩ {x ∈ eSe | ∃y ∈ S: yx = e}.

(Note that for separately perfect semigroups S the sets eSe are closed.)

(iii) A locally compact semigroup S is c-perfect if and only if it is ‘locallyperfect’—in the sense that for every point x there exists a neighborhood U suchthat the restriction of µ to (S×U)∪ (U ×S) is perfect. For semigroups definedon k-spaces the above concept of c-perfectness seems to be more adequate thanthat of local perfectness.

(iv) Suppose that in a locally compact c-perfect semigroup a product anbnconverges. Then the convergence of one of the factors implies that the otherhas a convergent subnet. Equivalently: if one of the factors has no convergentsubnet then the other has no convergent subnet either. (The straightforwardproof is left to the reader.)


(v) In a locally compact c-perfect semigroup every maximal subgroupH(e) is closed. (This follows from (iv) and the formula for H(e) in (ii).)

2.6. Examples. (i) Every compact topological semigroup is perfect.

(ii) If S is a compact topological semigroup and I is a closed ideal inS then the complement S\I of I is perfect if it is a semigroup. In fact, everylocally compact perfect semigroup can be so obtained (cf. 2.8 below).

(iii) If S is a topological semigroup with identity and the group of unitsis not compact then S cannot be perfect (by 2.4(ii)).

(iv) Similarly, a closed subsemigroup S of a topological group G cannotbe perfect if there exists a closed subgroup G1 of G such that S contains atranslate of the form [sG1 ] [G1s ] and for some t ∈ S the set [G1t∩S ] [tG1∩S ]is not compact (cf. also 4.3(i) and 7.8 below).

(v) Any closed subsemigroup S of a topological group G is c-perfect.Indeed, let K be a compact subset of S and write inv for the inversion ofG . Then for every s ∈ S the set {(x, y) ∈ K × S | xy = s} ∪ {(x, y) ∈S × K | xy = s} is compact, since it is closed and contained in the compactset K × inv(K)s∪ s inv(K)×K . Since the complex product AB of a compactsubset A with a closed subset B of G is always closed this implies that therestriction of the multiplication to (K × S) ∪ (S ×K) is perfect.

(vi) Every (left and right) cancellative semitopological semigroup withclosed (left and right) translations is separately perfect. In particular, a semi-topological semigroup is separately perfect if it is algebraically a group.

(vii) If S is a noncompact semitopological semigroup containing a com-pact ideal then S cannot be separately perfect. For instance, the multiplicativesemigroup S = [0,∞[ is not separately perfect.

(viii) The bicyclic semigroup, when endowed with the discrete topology,is c-perfect but not perfect.

2.7. Notation. Let S be a locally compact semigroup. Then we denote byS∞ = S∪{∞} the one point compactification of S , considered as the semigroupcompactification where the added point ∞ acts as a zero element. Note that ifS is compact then ∞ is an isolated point of S∞ .

2.8. Perfect semigroups and one-point compactifications (cf., e.g., [22])

For a locally compact topological semigroup S the following conditions are equiv-alent:

(i) S is a perfect semigroup.

(ii) If the product 〈sntn〉 of two nets converges in S then, upon passage to asuitable subnet, both 〈sn〉 and 〈tn〉 converge in S .

(iii) The one point compactification S∞ is a topological semigroup.


2.9. One point compactifications of c-perfect and of separately perfectsemigroups

The following assertions can be checked easily:

(i) A locally compact topological semigroup S is c-perfect if and only if themultiplication of S∞ is continuous at all points except possibly (∞,∞) .

(ii) A locally compact semitopological semigroup S is separately perfect if andonly if S∞ is a semitopological semigroup.

2.10. Under favorable circumstances c-perfectness can be deduced from sep-arate perfectness.

2.11. Proposition. Let S be a second countable locally compact separatelyperfect topological semigroup such that the sets sS and Ss , for every s ∈ S , areof second category. Then S is c-perfect.

Proof. By 3.5.11 and 4.2.8 of [6] the one point compactification S∞ ismetrizable. Denote by m the multiplication of S∞ . By Corollary 1.3 of [20],p. 49, there exists a residual set R ⊆ S such that m is continuous at each point(x, y) ∈ R × S∞ . By 2.9(i) we only have to show that m is continuous at allpoints (s,∞) and (∞, s) with s ∈ S . By symmetry it suffices to show thecontinuity at the points (s,∞).

Suppose that there are nets 〈sn〉 and 〈tn〉 in S such that the limitslim sn = s and lim sntn = x exist and are contained in S , but lim tn = ∞ .Then, since R is residual and Ss is of second category, we find an element a ∈ Swith as ∈ R . The continuity of m at (as,∞) implies that m(asn, tn) → ∞ .But lim a ·m(sn, tn) = ax ∈ S , a contradiction. Thus m is continuous at (s,∞)and the proof is finished.

An appropriate discussion of this subject in full generality would gobeyond the scope of the present paper.

2.12. A note on the terminology

(i) In the terminology of Neeb [19] locally compact perfect semigroups arecalled “proper semigroups.” Since this usage leads to terminological collisionswhenever proper subsets are involved (such as “proper proper subsemigroup”)we prefer to use the term “perfect.” Also, in the last decades the above definitionof perfect maps, which slightly deviates from Bourbaki’s original concept, hasproved to be very fruitful (also for spaces more general than k-spaces) andis commonly used among topologists. Perfect maps have good “preservation”properties, in fact, there is a fairly long list of topological cardinalities preservedfor images or inverse images under perfect maps. Also, some well knownproperties of compact topological semigroups, such as the Swelling Lemma,carry over to perfect or, mutatis mutandis, c-perfect semigroups on arbitraryspaces or on k-spaces. Thus we speculate that the concept of perfect semigroups


might be of interest also beyond the context of semigroups on locally compact(or k-) spaces.

(ii) Compact semigroups whose multiplication is jointly continuous exceptpossibly at a single point (z, z), where z is a zero element, have been consideredalready by K. H. Hofmann in the 60’s [12] (‘fasttopologische Halbgruppen’).

2.13. We now list several assertions about c-perfect or perfect semigroupswhich are familiar for compact topological semigroups, and in the case of locallycompact topological perfect semigroups are obvious consequences of the compactcase (by applying 2.8).

2.14. Continuity of the inversion. Let S be a topological semigroup de-fined on a k-space. Then S is a topological group if and only if it is algebraicallya group and c-perfect.

Proof. In view of Example 2.4(v) it suffices to prove the “if” part of theassertion. Suppose that K is a compact subset of a c-perfect topologicalsemigroup S which is defined on a k-space. Then, since S is c-perfect, theset {(x, y) ∈ K × S | xy = 1} = {(k, k−1) | k ∈ K} and hence the inverses k−1 ,k ∈ K , form a compact set. In a k-space a subset A is closed if and only ifA ∩K is closed for every compact subset K , so an involution on a k-space iscontinuous if and only if it sends compact sets to compact sets. This finishesthe proof.

2.15. Remark. The above discussion shows that c-perfectness can be con-sidered as a property of semigroups generalizing continuity of the inversion ingroups. If G is a group with separately continuous multiplication then it is sep-arately perfect, but by 2.14 it cannot be c-perfect if inversion is not continuous.For instance, the group of reals with the Sorgenfrey topology is a topologicalsemigroup on a k-space (note that this topology is first countable), but inversionis not continuous. Thus even for topological semigroups on k-spaces separateperfectness does not imply c-perfectness.

(More examples of groups on k-spaces with discontinuous inverse can befound in [16], [15], [20], the case of pseudocompact groups is treated in [9].)

2.16. Recall that by a famous theorem due to A. Weil every monotheticlocally compact topological group is either compact or topologically isomorphicwith Z . The following discussion shows that this result extends readily to c-perfect semigroups.

2.17. Monothetic c-perfect semigroups. Let S be a c-perfect semigroupon a nondiscrete locally compact space and suppose that there exists an elements ∈ S such that the powers sn , n ∈ N , are dense in S . Then S is compact.


Proof. Note first that S is the closure of an abelian semigroup, henceabelian. As S is not discrete, the set

Γ(s) =⋂m∈N

{sn | n ≥ m}

is nonvoid, and it follows from its definition that Γ(s) is a closed ideal of S .We prove that Γ(s) is a group.

Let x ∈ Γ(s) and let �,m ∈ N with � < m . Choose a compactneighborhood U of x and let 〈nα〉 be a net in m + N with x = lim snα andsnα ∈ U . By assumption the translation λs� is a perfect map, so we concludethat the elements snα−� are contained in a compact subset, namely λ−1

s�(U).

Passing to a suitable subnet we enforce that the limit t = lim snα−� exists. Itfollows that x ∈ s�{sn | n ≥ m− �} , for every m > � . Since the translationsin S are perfect the nonvoid set Am = {y ∈ {sn | n ≥ m− �} | s�y = x}is compact, for every m > � . Since the sets Am have the finite intersectionproperty their intersection

⋂m>�Am is not empty. But this intersection is

contained in Γ(s) so we have shown that Γ(s) = s�Γ(s), for every � ∈ N .

Now pick any y ∈ Γ(s). Then for every index α we find an elementtα ∈ Γ(s) with y = snαtα . Since S is c-perfect and since the elements snα liein the compact set U we conclude that the tα ’s also belong to a compact subset.Passing to a subnet we now enforce that tα converges. By the continuity of themultiplication this means that y ∈ xΓ(s). Thus xΓ(s) = Γ(s) and we concludethat Γ(s) is a group.

Let e be the identity of Γ(s). Then e{sn | n ∈ N} is dense in Γ(s),so Γ(s) is monothetic and the map Z → Γ(s), z �→ (es)z , is not a topologicalembedding (since the elements esz cluster at e). By the result of A. Weil citedabove this implies that Γ(s) is compact. Now the c-perfectness of S impliesthat the restriction of the multiplication to S × Γ(s) is perfect, so S must becompact.

2.18. For the following statement we recall that an element s in a semigroupS is said to swell a subset X of S if X is a proper subset of either sX or Xs .

2.19. The Swelling Lemma for c-perfect and perfect semigroups

Let S be a semigroup.

(i) If S is c-perfect then no element in S can swell a compact set.

(ii) If S is perfect then no element in S can swell a closed set.

Proof. It suffices to consider only the case X ⊆ sX , since the dual of aperfect [c-perfect] semigroup is perfect [c-perfect].

We first observe that X ⊆ sX implies X = sX whenever the setU = {u ∈ S | X ⊆ uX } is compact. In fact, if U is compact then so is the


closed semigroup P generated by the powers sn of s . (Note that U containsthe powers sn of s , by iteration.) In particular, the minimal ideal M(P ) of Pis a group. Let e be the identity of M(P ). Since e ∈ U we have X ⊆ eX ,hence eX = X . But we also have (es)−1 ∈ U , so sX ⊆ (se)(es)−1X = eX = Xand therefore X = sX .

Now we note that, in both cases, the set {(u, v) ∈ S × X | uv = x} iscompact and hence U = {u ∈ S | X ⊆ uX } is compact.

3. Compactly divided elements

3.1. By definition a topological semigroup S is perfect if (i) its multiplicationis a closed map and (ii) for every point s ∈ S the set {(x, y) ∈ S×S | xy = s} iscompact. In many situations of interest (i) is automatically satisfied if (ii) holds.To see this a closer look at condition (ii) is necessary. It is therefore natural tosingle out and study, in arbitrary topological semigroups, those elements whoseinverse image under the multiplication is compact. This is what we do in thepresent section.

3.2. Compactly divided elements

An element s in a topological semigroup S is called compactly divided if{(x, y) ∈ S × S | xy = s} is compact. We denote the set of compactly di-vided elements of S by cpdiv(S).

3.3. Divisors of an element

For an element s in a semigroup S we define

(i) the set LD(s) = {x ∈ S | s ∈ xS ∪ {x}} of left divisors of s in S ;

(ii) the set RD(s) = {x ∈ S | s ∈ Sx ∪ {x}} of right divisors of s in S ;

(iii) the set MD(s) = {x ∈ S | s ∈ xS ∪ SxS ∪ Sx ∪ {x}} of all divisors of s inS .

3.4. Proposition. Let S be a semigroup and s ∈ S . Then [S\LD(s)][S\RD(s)] [S\MD(s)] is the union of all [right ideals ] [ left ideals ] [ two-sidedideals ] of S which do not contain s . (Each of these sets may be empty.)

Proof. For notational simplicity, we suppose, without losing generality, thatS contains an identity. Consider an element x ∈ S\LD(s), i.e., s /∈ xS . Thens /∈ (xS )S either, hence xS ⊆ S\LD(s). Thus S\LD(s) is a right ideal ofS . If, conversely, R is a right ideal of S meeting LD(s) then s ∈ R , sinces ∈ xS ⊆ R for any x ∈ R ∩ LD(s). This proves the assertion for S\LD(s). Inthe other cases the assertion follows similarly.

3.5. Remark. (i) If S is a submonoid of a group then clearly LD(s) =S ∩ sS−1 , RD(s) = S ∩ S−1s , and MD(s) = S ∩ S−1sS−1 . In particular,LD(1) = RD(1) = MD(1) and LD(1) is the group of units H(S) of S .


(ii) It is not difficult to see that the elements in [LD(1)\RD(1)] [RD(1)\LD(1)] are the [left] [right] magnifyers in the sense of the algebraic theory ofsemigroups (Lyapin [14], p. 118). Thus the equation LD(1) = RD(1) (whichimplies LD(1) = MD(1)) just means that S does not contain magnifyers.

(iii) For every s in a locally compact c-perfect semigroup the sets LD(s)and RD(s) are closed, by 2.4(iv). In a perfect semigroup the sets LD(s) andRD(s) are always compact.

3.6. Connectedness of the divisor sets

Let S be a topological monoid.

(i) Assume that ϕ1, ϕ2, . . . , ϕk are one parameter semigroups R+0 →

S . Then for every choice of parameters t1, t2, . . . , tk ∈ R+0 the element s =

ϕ1(t1)ϕ(t2) · · ·ϕk(tk) and the identity are contained in the same path connectedcomponent of [LD(s)] [RD(s)] [MD(s)].

Indeed (cf. [11], V.2.6): Put T0 = 0 and Tj = t1 + t2 + · · · tj , forj = 1, 2, . . . , k . Then the curve α: [0, Tk] → LD(s), defined by α(0) = 1and

α(t) = ϕ1(t1) · · ·ϕj(tj)ϕj+1(t− Tj) for Tj < t ≤ Tj+1, j = 0, 1, . . . , k − 1,

is continuous and joins 1 with s . The cases of RD(s) and MD(s) are left tothe reader.

(ii) If S is generated by its one parameter subsemigroups, i.e., everyelement s ∈ S has a product decomposition as in (i), then for each s ∈ S thethree sets LD(s), RD(s) and MD(s) are path connected. This follows from (i)and the transitivity of the divisor relations: s′ ∈ LD(s) =⇒ LD(s′) ⊆ LD(s),and s′ ∈ RD(s) =⇒ RD(s′) ⊆ RD(s).

(iii) The set A = {a ∈ S | a and 1 lie in the same [connected] [pathconnected] component of LD(a)} is a subsemigroup of S .

In fact, let a, b ∈ A and choose [connected] [path connected] subsets Ca

in LD(a), and Cb in LD(b) with {1, a} ⊆ Ca , {1, b} ⊆ Cb . Then the setCa ∪ aCb is [connected] [path connected] and joins 1 with ab .

3.7. Proposition. For an element s of a c-perfect semigroup S the follow-ing assertions (i) and (ii) are equivalent:

(i) One of the sets LD(s) , RD(s) , MD(s) has compact closure.

(ii) Each of the sets LD(s) , RD(s) , MD(s) is compact.

If the multiplication of S is jointly continuous then these assertions are alsoequivalent with

(iii) s is compactly divided.


Proof. (i) =⇒ (ii) For simplicity we assume that S contains an identity,this will inflict no harm to generality. Since MD(s) contains LD(s) ∪ RD(s),and since RD(s) is carried to LD(s) if we pass to the reverse semigroup, itsuffices to show that the three divisor sets are compact if LD(s) has compactclosure.

Suppose that LD(s) has compact closure. Since S is c-perfect therestriction of the multiplication to the product LD(s) × S is perfect, thus theset

{(x, y) ∈ S × S | xy = s} = {(x, y) ∈ LD(s)× S | xy = s}

is compact and it follows that LD(s) and RD(s) must be compact. By 2.3(ii)the set

L(s) = {(x, y) ∈ LD(s)× S | xy ∈ LD(s)}

is compact. The composition of perfect maps is perfect, so the map

L(s)× S → S, ((x, y), z) �→ xyz ,

which is the composition of the perfect maps L(s)×S → LD(s)×S , ((x, y), z) �→(xy , z), and LD(s) × S → S , (u, v) �→ uv , is perfect. We conclude that theset {(x, y, z) ∈ S × S × S | xyz = s} must be compact, which implies thecompactness of MD(s).

It is obvious that (ii) =⇒ (i). If µ is continuous then, since µ−1(s) ⊆LD(s)×RD(s) and since LD(s) is a continuous image of µ−1(s), we also have(ii) =⇒ (iii) =⇒ (i).

3.8. An application to the group of units. For a c-perfect monoid Sthe following assertions are equivalent:

(i) The identity 1 of S is compactly divided.

(ii) H(S) = LD(1) = RD(1) = MD(1) (equivalently: S contains no magni-fyers) and H(S) is compact.

(iii) H(S) is compact and a face of S , i.e., ab ∈ H(S) implies a, b ∈ H(S) .

Proof. (i) =⇒ (ii) We know from 3.7 that the divisor sets LD(1), RD(1),and MD(1) are compact. Pick a, b ∈ S with ab = 1. Then bRD(1) ⊆ RD(1)(since RD(1) is a semigroup), so bRD(1) ⊆ abRD(1). In a c-perfect semigroupno element can swell a compact set (by the Swelling Lemma 2.19, so we concludethat bRD(1) = RD(1). This means that b ∈ LD(1), and a ∈ RD(1) (notethat bc = 1 implies c = abc = a). Thus LD(1) = RD(1) = H(S). The prooffor the equality MD(1) = H(S) is now straightforward.

The implications (ii) =⇒ (iii) =⇒ (i) are left to the reader.


3.9. A remark on compact order intervals

Recall that every subsemigroup S of a semigroup G induces a natural leftinvariant pre-order ≤ on G , defined by x ≤ y if y ∈ xS ∪ {x} . If x ≤ ythen the order interval [x, y] is defined to be the set {s ∈ G | x ≤ s ≤ y} .For applications in the harmonic analysis of semisimple Lie groups (cf., e.g.,Faraut [7], [8], Neeb [17], [18]) it is important to know whether for someclosed subsemigroup S the order intervals [x, y] are compact or not. If Gis a topological group then [x, y] = {x} ∪ xLD(u), with y = xu , u ∈ S , and Sis c-perfect, so [x, y] is compact if and only if u is compactly divided.

4. Aliens

4.1. Notation. In the following we shall write limxn =∞ if 〈xn〉 is a netin a locally compact space which is eventually outside of every compact subset.In other words, limxn =∞ means that 〈xn〉 has no convergent subnet.

4.2. Aliens

Let S be a locally compact semigroup. An element s ∈ S is called an alienelement, or an alien for short, if it is the limit of “products with arbitrarilylarge (left or right) factors”, that is to say, s is the limit of a net 〈unvn〉 suchthat either limun = ∞ or lim vn = ∞ (or both). The set of aliens in Swill be denoted by Al(S). An element s ∈ S is called a pure alien, if it has“arbitrarily large factors”, i.e., there are two nets 〈un〉 and 〈vn〉 , such thateither limun = ∞ or lim vn = ∞ , and such that s = unvn for every index n .Equivalently, s is a pure alien if it is not compactly divided. The set of purealiens will be denoted by PAl(S). Obviously, PAl(S) ⊆ Al(S), so all elementsoutside Al(S) are compactly divided.

4.3. Elementary properties of aliens

In the following assertions S always denotes a locally compact semigroup.Proofs are left to the reader.

(i) For a locally compact topological semigroup S the following assertionsare equivalent (by 2.8): (a) S is perfect, (b) Al(S) = ∅ , (c) multiplication isclosed and PAl(S) = ∅ .

(ii) If the group of units H(S) of a locally compact topological monoidS is not compact then S = PAl(S) = Al(S). (Note that if the closure ofH(S) is compact then so is H(S).) We shall see in 7.5(iii) and 7.6(iv) thatthere are locally compact topological semigroups S without identity such thatS = Al(S).

(iii) Let f : S → T be a homomorphism of locally compact semigroupswhich is a perfect map. Then f(Al(S)) ⊆ Al(T ).

(iv) If T is a dense subset of S then for every alien s ∈ Al(S) we canchoose nets 〈un〉 and 〈vn〉 in T such that s = limunvn and either limun =∞or lim vn =∞ .


(v) The set Al(S) is closed, in fact:

Al(S) =⋂{(S\K)S ∪ S(S\K) | K ⊆ S,K compact}.

If S is locally compact and c-perfect then by 2.4(iv) we have the slightly simplerformula

Al(S) =⋂{(S\K)(S\K) | K ⊆ S, Kcompact}.

(vi) If A,B are closed subsets of a locally compact topological semigroupthen the set AB ∪Al(S) is closed. If every alien of a semigroup S is pure, thenS2 ⊇ Al(S) = PAl(S), so S2 is closed in S .

4.4. Examples. (i) Every locally compact topological semigroup A canoccur as the set of aliens in some locally compact topological semigroup S .To see this, let S be the (disjoint) topological sum of the spaces N and A . Weextend the given multiplication of A to a multiplication ∗ on S by stipulatinga ∗ b = ab , m ∗ n = m+ n and a ∗m = m ∗ a = a for any a, b ∈ A , m,n ∈ N .Then PAl(S) = Al(S) = A . (Note that S is not separately perfect.)

(ii) Let S be a noncompact locally compact topological space and z ∈ S .Then PAl(S) = Al(S) = {z} if S is provided with the constant multiplicationxy = z , for all x, y ∈ S .

4.5. Aliens in separately perfect semigroups

Suppose that S is a locally compact and separately perfect semigroup. Thenthe following assertions can be checked readily.

(i) Each of the sets PAl(S) and Al(S) is either empty or an ideal in S .

(ii) If I is a closed ideal of S then Al(S) �= ∅ if and only if Al(I) �= ∅ .

(iii) If A1, A2, . . . , Ak are closed subsets of S , and if S is a topological semi-group then the set A1A2 · · ·Ak∪Al(S) is closed. (Apply 4.3(v), the aboveassertion (i), and induction.)

4.6. The Tube Theorem for nonaliens. Let S be a noncompact locallycompact c-perfect semigroup and s ∈ S\Al(S) . Then the complements S\I ofclosed [ ideals ] [right ideals ] [ left ideals ] I not containing s form a neighbor-hood basis of the compact set [MD(s)] [LD(s)] [RD(s)] .

Proof. We only prove the first alternative, the others are left to the reader.Since PAl(S) ⊆ Al(S) we know that s is compactly divided, so, by 3.7, MD(s)is compact and �= S . Let U �= S be an open neighborhood of MD(s). Thenthe set

I = S(S\U)S ∪ S(S\U) ∪ (S\U)S ∪ (S\U) ∪Al(S)

is a closed ideal, by 4.5(i) and (iii). Also, s /∈ I . Thus S\I is an openneighborhood of MD(s) (by 3.4) and contained in U . It is plain that if I is anideal not containing s then it cannot intersect MD(s).


4.7. Corollary.

(i) Let S be a noncompact locally compact c-perfect monoid. If 1 /∈ Al(S)then H(S) is compact and the complements S\I of closed proper ideals Iin S form a neighborhood basis for H(S) .

(ii) Let S be a closed noncompact submonoid of a locally compact topologicalgroup. If Al(S) = PAl(S) and H(S) is compact then the complementsS\I of closed proper ideals I in S form a neighborhood basis for H(S) .

Proof. (i) If 1 is not an alien then it is compactly divided, hence H(1) iscompact. The rest of the assertion follows from 3.8 and 4.6.

(ii) Since S is a subset of a group we conclude that LD(1) = RD(1) =H(S), since S is c-perfect this implies that 1 is compactly divided (by 3.7).Thus H(S) ∩Al(S) = ∅ and the assertion follows from (i).

4.8. For our next result we need a little lemma about the interior of ideals ofsubsemigroups of topological groups.

4.9. Lemma. Let G be a topological group.

(i) If A is a subset of G with 1 ∈ A and O is open in G then O ⊆ OA.

(ii) If S is a subsemigroup of G with 1 ∈ Int(S) then every ideal I of Ssatisfies Int(I) = Int(I)S = Int(I) Int(S) = I Int(S) .

Proof. (i) Consider an arbitrary o ∈ O . Then O−1o is an open set contain-ing 1 , hence O−1o ∩A �= ∅ and therefore o ∈ OA .

(ii) The inclusion Int(I) ⊆ Int(I) Int(S) follows from (i). Since Int(I)Int(S) ⊆ I Int(S) ⊆ Int(I) and Int(I)S ⊆ Int(I) the equalities of (ii) follow.

The following proposition is essentially a translation of Lemma V.7 inNeeb’s memoir [19] into our terminology and notation.

4.10. Proposition. Let S be a closed submonoid of a locally compact topo-logical group G and suppose that S has dense interior in G . Then Al(S) hasdense interior in G and Int(Al(S)) ⊆ PAl(S) . In particular, PAl(S) = Al(S) ,so S is perfect if and only if all of its elements are compactly divided.

Proof. Since in a topological group the translations carry open sets ontoopen sets and since S has inner points in every neighborhood of the identitywe see that every left or right ideal of S , in particular Al(S), has denseinterior in G . Pick s ∈ Al(S) and u ∈ Int(S). Let s = lim anbn withlim an = lim bn = ∞ in S . Since u is an interior point of S the element(anbn)

−1su lies in S whenever n is sufficiently large. Thus su = unvn whereun = an and vn = bn(anbn)

−1su = a−1n su . By 4.9(ii) we know that every

element of Int(Al(S)) is of the form su with s ∈ Al(S) and u ∈ Int(S), so


Int(Al(S)) ⊆ PAl(S). Thus Al(S) ⊆ PAl(S). Since PAl(S) ⊆ Al(S) and sinceAl(S) is closed (4.3(v)) the reverse inclusion holds as well and the proof iscomplete.

4.11. Theorem. Let S be a locally compact c-perfect semigroup and assumethat S is compactly generated, that is, there exists a compact set C ⊆ S suchthat the union D =

⋃{Cn | n ∈ N} is dense in S . Then the following assertions

hold:

(i) Every alien in S is a pure alien, PAl(S) = Al(S) .

(ii) The compactly divided elements form an open subset of S .

(iii) S is perfect if and only if every element is compactly divided.

Proof. (i) Pick an alien s ∈ Al(S). We show that for every compact subsetK of S we have s ∈ (S\K)S , so that s must be a pure alien.

Choose an arbitrary compact subset K of S , and let U be a relativelycompact open neighborhood of K ∪ C . Since s is an alien (and since S is c-perfect) there exist nets 〈un〉 and 〈vn〉 in D\U such that limun = lim vn =∞and limunvn = s .

We claim that every un has a left divisor u′n with u′n ∈ UC\U . Indeed,we may write un = c1c2 · · · ck , where the ci ’s are taken from C . By ourchoice of U we have c1 ∈ U . Let j be the largest index such that the product

c1c2 · · · cj ∈ U . Then u′ndef= c1c2 · · · cj+1 ∈ UC\U . Obviously, u′n is a left

divisor of un .

By passage to a suitable subnet we enforce that the limit u = limu′n inthe compact set UC\U exists. Then, since S is locally compact and c-perfect,u must be a left divisor of s . Clearly, u /∈ K and the proof of (i) is finished.

Assertions (ii) and (iii) are immediate consequences of (i).

Lie semigroups are compactly generated, thus the following corollary isan immediate consequence of 4.11.

4.12. Corollary. ([19], Theorem V.9(i)) In a Lie semigroup S every alienis pure and the set cpdiv(S) of compactly divided elements of S is open in S .

4.13. Corollary. If S is a compactly generated locally compact c-perfectsemigroup, in particular: if S is a closed locally compact and compactly gener-ated subsemigroup of a topological group, then S2 is closed.

Proof. By 4.11, we know that PAl(S) = Al(S), so the assertion follows from4.3(vi).

Assertion (iii) of the next corollary motivates the investigation of “largeideals” in the next section. For Lie semigroups S the implication (ii) =⇒ (iii)of the corollary is the well known Tube Theorem for compact unit groups ([11],p. 389).


4.14. Corollary. Let S be a noncompact closed locally compact submonoidof a topological group and assume that S is compactly generated. Then thefollowing assertions are equivalent:

(i) Al(S) �= S .

(ii) The group of units H(S) of S is compact.

(iii) H(S) has a neighborhood basis consisting of complements of proper co-compact ideals in S .

Proof. (i) ⇐⇒ (ii) By 4.11 we know PAl(S) = Al(S). Since 1 ∈ PAl(S)if and only if LD(1) = H(S) is not compact, this implies the assertion.(ii) =⇒ (iii) follows from 4.7(ii) and 4.11. The implication (iii) =⇒ (ii) is triv-ial since H(S) is closed.

4.15. Remark. The arguments in the proof of 4.11 can be adapted to workalso under a connectedness condition:

Suppose that in a locally compact c-perfect monoid S the set

A = {a ∈ S | a and 1 lie in the same connected component of LD(a)}

is dense in S . Then every alien in S is pure.

In this case we pick K and s as in the proof of 4.11. Let U be a relativelycompact open neighborhood of K ∪ {1} . Since s is an alien there exist nets〈un〉 and 〈vn〉 in A\U such that limun = lim vn =∞ and limunvn = s .

We claim that every un has a left divisor u′n which lies in ∂U . Byassumption, 1 and un lie in the same connected component of LD(un). Thiscomponent contains a point inside U , namely, 1 , and a point outside U , namely,un . Hence there exists an element u′n ∈ ∂U ⊆ U such that u′n ∈ LD(un).

Since ∂U is compact we can assume that u = limu′n exists in ∂U . Then,since S is locally compact c-perfect, u ∈ LD(s). Clearly, u /∈ K , and the proofis complete.

However the only known case where the above connectedness conditioncan be checked comfortably is when S contains a dense subsemigroup generatedby one parameter semigroups (3.6(ii)). But then S is generated by everyneighborhood of the identity (hence is compactly generated), so this case isalready covered by 4.11.

4.16. Examples. In the following examples all semigroups will be subsemi-groups of the vector group R2 .

(i) A1 = [1,∞)× R . Here PAl(A1) = Al(A1) = A1 +A1 = [2,∞)× R .

(ii) The additive semigroup A2 = {(x, y) ∈ R2 | 0 ≤ y ≤ x} is a pointedcone, hence is a perfect semigroup, PAl(A2) = Al(A2) = ∅ .

(iii) A3 = A1 ∪ { 12} ×

(N ∪ −N

√2). In this semigroup the pure aliens

are not dense in the set of all aliens. In fact, PAl(A3) = [1.5,∞) × R but


Al(A3) = PAl(A3)∪ {1} ×R . (Recall that the inequality |m− n√2| < 1/n has

infinitely many solutions (m,n) ∈ N×N , so the semigroup {m−n√2 | m,n ∈ N}

is dense in R .)

(iv) A4 = ({(0, 0)} ∪A3)+([0,∞)× {0}). This semigroup is a connectedsubmonoid of R2 , but the pure aliens are not dense in the set of all aliens. Wehave PAl(A4) = [1.5,∞)× R and Al(A4) = A1 .

(v) A5 = (A3 + A2) ∪ A2 . This semigroup is a submonoid of R2 withdense interior, but the pure aliens are properly contained in the set of all aliens.We have PAl(A5) = (1,∞)× R and Al(A5) = A1 .

(vi) A6 = S1 ∪ S2 . The semigroup A6 is a submonoid of R2 with denseinterior, A1 is a closed ideal in A6 with relatively compact complement andPAl(A6) = Al(A6) = A1 + A1 . However, for s ∈ A6\(A2 ∪ A1 + A1) theelements s and 1 do not belong to the same connected component of LD(s).

Note that the semigroups A3 , A4 , A5 and A6 are not compactly gener-ated (and do not satisfy the connectedness condition of the previous remark).

5. Large ideals

5.1. Large sets in topological spaces

A subset A of a topological space X is called large, if X\A is relatively compact.Equivalently: A is large if every net 〈xn〉 in X with xn → ∞ eventually liesin A .

5.2. Theorem. Let S be a noncompact locally compact c-perfect semigroup.Then

Al(S) =⋂{L ⊆ S | L is a large closed left ideal in S}

=⋂{R ⊆ S | R is a large closed right ideal in S}

=⋂{I ⊆ S | I is a large closed ideal in S}

=⋂{T ⊆ S | T is a large closed subsemigroup of S}.

In particular, the intersection of all closed large ideals is empty if S is perfect.

Proof. We first pick a large closed subsemigroup T of S and show thatT contains Al(S). Let s ∈ Al(S) and consider nets 〈un〉 and 〈vn〉 suchthat limun = lim vn = ∞ and s = limunvn . Then the elements un and vneventually lie in T . Since T is a closed semigroup this implies s = limunvn ∈ T .

It now suffices to show that no element of S\Al(S) can sit in all closedlarge ideals. Pick a point s ∈ S\Al(S) and choose a compact neighborhood Uof the compact set MD(s). Then by the Tube Theorem 4.6 there exists a closedideal I of S , not containing s and such that S\I ⊆ U . Since U is compact wethus have found a closed large ideal not containing s , which finishes the proof.


5.3. Rees quotients of topological semigroups modulo large ideals

Let I be a large closed ideal in a locally compact topological semigroup S andlet q: S → S/I be the canonical quotient morphism onto the Rees quotientS/I . We write ω for the zero element in S/I (so q(I) = {ω}).

(i) If 〈sn〉 is a net in S converging to ∞ then q(sn) = ω for all sufficientlylarge indexes n , so lim q(sn) = ω .

(ii) S/I is a compact topological semigroup.

Proof. (i) Since I is large we have sn ∈ I and hence q(sn) = ω for allsufficiently large indexes n .

(ii) Applying elementary topology (cf., e.g., [2] I,§8.6) we conclude thatS/I is a Hausdorff space. From q(S) = q(S\I) ∪ q(I) we deduce that S/I =q(S\I) ∪ {ω} , hence S/I is compact. The proof of the continuity of themultiplication of S/I is left to the reader.

5.4. Projective limits. (see [5], p. 80 ff.)

We recall that if S is a topological semigroup, {Ri} is a family of closedcongruences on S directed under ⊇ , such that each S/Ri is a topologicalsemigroup, then the family {S/Ri} can be considered as a (strict) projective(inverse) system of topological semigroups, in a natural manner.

5.5. The large ideal compactification

Let S be a noncompact locally compact topological semigroup. Then the setI of all large closed ideals of S is directed under ⊇ since for I, J ∈ I theintersection I ∩ J also lies in I . Thus, by 5.3(ii) and 5.4, {S/I}I∈I forms aprojective system of compact topological semigroups. The limit of this systemexists and is a compact semigroup (cf. Theorem 2.22 of [5]). We write S� for thissemigroup and i� for the projective limit S → S� of the quotient morphismsS → S/I . The pair (S�, i�), or, if no ambiguity is to be feared, S� , is called thelarge ideal compactification of S . (As usual, this compactification is identifiedwith all equivalent ones.)

5.6. An explicit construction of S� . For every I ∈ I denote by qI : S →S/I the canonical quotient map. Let qI(I) = {ωI} . Define the continuous mapϕ: S → ΠI∈IS/I by ϕ(s) = (qI(s))I∈I . Then the following assertions hold:

(i) The space ϕ(S) , together with the corestriction ϕ∗ of ϕ to a map S →ϕ(S) , is a representative of the large ideal compactification.

(ii) ϕ(S) = ϕ(S) ∪ {(ωI)I∈I} .

(iii) If⋂

I∈I I �= ∅ then ϕ(S) = ϕ(S) .


(iv) The restriction of ϕ to the subspace S\⋂

I∈I I is a topological embedding.Thus if

⋂I∈I I = ∅ then the large ideal compactification is equivalent to

the Alexandrov compactification.

Proof. (i) The inclusion ϕ(S) ⊆ lim←−

SI follows from the definition of the

projective limit. The assertion follows now from 2.5.6 of [6].

(ii) Let 〈sn〉 be a net in S such that 〈ϕ(sn)〉 converges to some (aI)I∈I ∈ϕ(S). If 〈sn〉 has a convergent subnet then clearly (aI)I∈I ∈ ϕ(S). If sn →∞then aI = ωI for every I ∈ I , by 5.3(i). Thus ϕ(S) ⊆ ϕ(S) ∪ {(ωI)I∈I} . Forthe converse inclusion consider a net 〈sn〉 in S converging to ∞ (such a netexists since S is not compact). Applying once again 5.3(i) we conclude that(ωI)I∈I = limϕ(sn) ∈ ϕ(S).

The proofs for (iii) and (iv) are left to the reader.

5.7. The large ideal compactification as a Rees quotient of S∞

The set A =⋂

I∈I I is the inverse image of the zero element (ωI)I∈I underthe compactification map ϕ∗ of 5.6, A is either empty or a closed ideal inS . Thus A ∪ {∞} is a closed ideal in S∞ . Let i: S → S∞ be the inclusionand q: S∞ → S∞/A ∪ {∞} be the canonical Rees quotient map. We endowthe semigroup S∞/A ∪ {∞} with the quotient topology, which is compact(Hausdorff), and put ψ = q◦i . By construction there exists an algebraicisomorphism of semigroups f : S∞/(A∪ {∞})→ ϕ(S) such that f◦ψ = ϕ . Weclaim that f is also a homeomorphism, so that S∞/(A ∪ {∞}) is a topologicalsemigroup which is (both algebraically and topologically) isomorphic with S� .

Proof. For the continuity of f it suffices to show that if s ∈ S and 〈sn〉 isa net in S such that limψ(sn) = ψ(s) then 〈ϕ(sn)〉 converges to ϕ(s). Sinceϕ(S) is compact we may suppose that 〈ϕ(sn)〉 is convergent.

Case 1: 〈sn〉 has a subnet converging in S to some s′ . Then ψ(s) =ψ(s′), so ϕ(s) = ϕ(s′). Hence 〈ϕ(sn)〉 converges to ϕ(s′) = ϕ(s).

Case 2: lim sn = ∞ . The equality ψ(s) = limψ(sn) then implies thats ∈ A . Thus ϕ(s) = (ωI)I∈I . On the other hand, limϕ(sn) = (ωI)I∈I ,by 5.3(i). Hence limϕ(sn) = ϕ(s) and the continuity of f follows. SinceS∞/A ∪ {∞} is compact and ϕ(S) is Hausdorff we conclude that f is ahomeomorphism.

Combined with the characterization of Al(S) as the intersection of largeideals in 5.2 the above observations yield:

5.8. Theorem. Let S be a noncompact locally compact c-perfect semigroup.Then S∞/(Al(S) ∪ {∞}) is a compact topological semigroup and the ensuingcompactification is equivalent with the large ideal compactification.

5.9. Remark. (i) The above theorem was inspired by assertion (2) of The-orem V.8 in Neeb’s memoir [19], which states that S∞/(Al(S) ∪ {∞}) is a


compact topological semigroup if S is a closed subsemigroup of a locally com-pact topological group with 1 ∈ IntS .

(ii) Theorem 5.8 shows that the large ideal compactification of a noncom-pact locally compact c-perfect semigroup S is isomorphic with the one pointcompactification S∞ if and only if S is perfect.

6. Alien elements in subsemigroups of Sl(2,R)+

6.1. Notation. We use the notation of [4], notably

• Sl(2,R)+ denotes the semigroup of all matrices in Sl(2,R) with nonneg-ative entries.

• H =(

1 00 −1

), P =

(0 10 0

), Q =

(0 01 0

).

• Let g =(a bc d

)∈ Sl(2,R)+ with a+ d > 2. Then

rlog(g)def=

(a−du

2bu

2cu −a−d

u

)with u =

√(a+ d)2 − 4.

We know that g = exp(log((a+ d+ u)/2) · rlog(g)).

• For any subsemigroup S of a Lie group G with Lie algebra g the umbrella

set Umb(S) is defined by Umb(S)def= {X ∈ g | ∃t0 > 0: exp(tX ) ∈

S, ∀t > t0}.

6.2. Alien pairs

For the purposes of this section we define an alien pair sequence, or an alienpair for short, to be a sequence 〈xn, yn〉 , with xn, yn ∈ Sl(2,R)+ and such thatxn →∞ , yn →∞ , whereas 〈xnyn〉 is bounded.

If the matrices xn, yn of an alien pair are taken from a fixed closedsubsemigroup S of Sl(2,R)+ then we say that S admits alien pairs.

The following assertions are immediate consequences of the definitions:

(i) Every subsequence of an alien pair sequence is an alien pair sequence.

(ii) In Sl(2,R)+ conjugation by the matrix j =(0 11 0

)(an outer automorphism

of Sl(2,R)) or taking transposes (an antiautomorphism) sends alien pairsto alien pairs.

(iii) A closed subsemigroup S of Sl(2,R)+ contains aliens if and only if itadmits alien pairs. Thus a closed subsemigroup S of Sl(2,R)+ is perfectif and only if it has no alien pairs.


6.3. Further basic properties of alien pairs

In the following we assume that 〈xn, yn〉 is an alien pair, and write

xn =

(an bn

cn dn

), yn =

(a′n b′nc′n d′n

).

(i) Since Sl(2,R)+ is closed in Gl(2,R), at least one of the entries of xn , andat least one of the entries of yn , must be unbounded.

(ii) By the definition of alien pairs and since all entries involved are nonneg-ative, the products

ana′n, bnc

′n, anb

′n, bnd

′n,

cna′n, dnc

′n, cnb

′n, dnd

′n,

form bounded sequences.

(iii) Either 〈an〉 or 〈dn〉 is unbounded. Similarly, taking transposes, either〈a′n〉 or 〈d′n〉 is unbounded.

Proof. Suppose first that, contrary to our claim, both of the entries anand dn are bounded, so that either bn or cn is unbounded. If bn isbounded then conjugation with the matrix j =

(0 11 0

)interchanges bn and

cn (as well as an and dn ), so we may assume that lim sup bn = ∞ .Passing to a suitable subsequence we enforce that bn →∞ . Now, by theboundedness of the products bnc

′n and bnd

′n in the first line of our list

in (ii), it follows that lim c′n = lim d′n = 0, and we conclude from (i) thateither 〈a′n〉 or 〈b′n〉 must be unbounded. Inspecting once more the firstline in our list in (ii) above, we see that this implies lim inf an = 0, and,as a consequence, lim inf andn = 0 (note that 〈dn〉 is bounded). Becauseof andn = 1 + bncn ≥ 1 this is a contradiction.

(iv) If lim sup an =∞ then, upon passing to a suitable subsequence, lim bn =lim dn = lim a′n = 0 .

Similarly, lim sup dn = ∞ implies that after passing to a subsequencelim an = lim cn = lim d′n = 0 .

Proof. Upon passing to a suitable subsequence and/or applying conju-gation by

(0 11 0

)it suffices to show the assertion under the assumption

an → ∞ . Then by (ii) a′n → 0 and b′n → 0, so either lim sup c′n = ∞or lim sup d′n = ∞ . In both cases our list in (ii) yields lim inf bn =lim inf dn = 0.

(v) Applying conjugation with j =(0 11 0

)and/or after passing to a suitable

subsequence, if necessary, we can always enforce that the following asser-tions hold:


(b) 0 < lim ana′n <∞ ,

(l) lim an = ∞ , lim bn = lim dn = 0 and the limits lim cn/an , lim bnanexist.

(r) lim d′n = ∞ , lim a′n = lim b′n = 0 and the limits lim c′na′n , lim b′n/a

′n

exist.

Proof. We start by passing to a suitable subsequence such that all productslisted in (ii) converge. Next, applying the above assertions (iii), (iv),and applying conjugation with j if necessary, we enforce that lim an =lim d′n = ∞ , lim bn = lim b′n = 0. Since andn = 1 + bncn ≥ 1 and,similarly, a′nd

′n ≥ 1 we have ana

′ndnd

′n ≥ 1, so lim ana

′n > 0. This implies

that lim cn/an = lim cna′n/ana

′n and, similarly, lim b′n/a

′n exist. Now we

observe that anbn ≤ a′nd′nanbn = ana

′nbnd

′n , thus 〈anbn〉 is bounded,

hence admits a convergent subsequence. A similar argument shows that〈c′na′n〉 admits a convergent subsequence, too.

(vi) If 〈xn〉 , 〈yn〉 are sequences in Sl(2,R)+ such that the conditions listed in(v) hold then 〈xn, yn〉 is an alien pair.

Proof. This is seen immediately from the decompositions

(an bn

cn dn

)=

(1 0

cn/an 1

)(1 anbn

0 1

)(an 0

0 1/an

)


)=

(a′n 0

0 1/a′n

)(1 0

c′na′n 1

)(1 b′n/a

′n

0 1

).

6.4. Normalized alien pairs

We henceforth call an alien pair 〈xn, yn〉 normalized if it satisfies the conditionsof 6.3(v). Note that every alien pair contains a subsequence 〈x∗n, y∗n〉 , such thateither 〈x∗n, y∗n〉 or its conjugate j〈x∗n, y∗n〉j = 〈jx∗nj, jy∗nj〉 is normalized.

6.5. Remark. Let S be a closed subsemigroup of Sl(2,R) and T a densesubset of it. If S admits normalized alien pairs, then it admits also normalizedalien pairs whose elements are chosen from T .

6.6. Proposition. In Sl(2,R)+ , let S be a closed subsemigroup, M a closedsubset, and assume that the diagonal entries of the matrices in M are ≥ 1 . Wedenote with SM the closed semigroup generated by S ∪M . Then SM admitsnormalized alien pairs if and only if S admits normalized alien pairs.

(Thus SM is a perfect semigroup if and only if S is a perfect semigroup.)

Proof. The “if” part being trivial, only the “only if” part has to be proved.Write S0

M for the set of all finite products z1z2 · · · zk , k ∈ N , where each of the


zi ’s either lies in S or in M . By definition, S0M is dense in SM . Suppose that

SM admits normalized alien pairs. Then by 6.4 we can find a normalized alienpair 〈xn, yn〉 in SM with xn, yn ∈ S0

M , that is, we have product decompositions

xn = z1z2 · · · zkn, yn = z′1z

′2 · · · z′ln ,

where each of the zi ’s either lies in S or in M . We form new sequences 〈x∗n〉and 〈y∗n〉 by replacing any factor zj , z

′j by the identity if it belongs to M . For

the entries of xn, yn and, mutatis mutandis, x∗n, y∗n we use the notation of 6.3.

For sufficiently large indexes n we have x∗n, y∗n ∈ S , since lim dn = lim a′n = 0.

Since the diagonal entries of the matrices in M are ≥ 1 we conclude that, entryby entry, x∗n ≤ xn , y

∗n ≤ yn . In particular, d∗n ≤ dn , so d∗n → 0 and therefore

a∗n → ∞ (since a∗n ≥ 1/d∗n ). Thus x∗n → ∞ . In the same vein, a′∗n → 0implies that y∗n →∞ . The entries of the product matrices xnyn are bounded,so 〈x∗n, y∗n〉 contains a subsequence which is a normalized alien pair in S .

6.7. Proposition. Suppose that S is a closed subsemigroup of Sl(2,R)+

with dense interior and which admits a normalized alien pair 〈xn, yn〉 . Thenthere exist nonnegative numbers γ , γ′ such that H + γQ ∈ Umb(S) and−H + γ′Q ∈ Umb(S) .

Obviously, if the conjugate j〈xn, yn〉j is normalized then there exist nonnegativenumbers β , β′ such that H + βP ∈ Umb(S) and −H + β′P ∈ Umb(S).

Proof. In view of 6.4 we may assume, without losing generality, that xn, yn ∈IntS . Then rlog(xn), rlog(yn) ∈ Umb(Int(S)). From 6.3(v) and the formulafor rlog we deduce that

rlog(xn) =

(αn βn

γn −αn

)with αn → 1, βn → 0, γn → γ = 2 lim

cnan,

and similarly,

rlog(yn) =

(α′n β′nγ′n −α′n

)with α′n → −1, β′n → 0, γ′n → γ′ = lim

c′na′n

c′na′n

b′na′n+ 1

.

Hence H + γQ,−H + γ′Q ∈ Umb(S).

6.8. Supplement. Let G be the Borel subgroup exp(RH + RX) , whereeither X = P or X = Q , and suppose that S is a closed subsemigroup ofexp(RH + R+

0 X) with dense interior in G . If S admits a normalized alienpair then there exist nonnegative numbers γ , γ′ with H + γQ ∈ Umb(S) and−H + γ′Q ∈ Umb(S) .

Note that for X = P we get γ = γ′ = 0, so that both H and −H lie inUmb(S).


Proof. The proof of 6.7 can be adapted easily to apply to this supplement,we only have to take the interior IntS with respect to G .

6.9. Corollary. Suppose that, in addition to the assumptions of Proposi-tion 6.7, S has connected interior. Then Q belongs to Umb(S) .

Proof. By Theorem 10.5 of [4] we know that Umb(S) is a semialgebra withinner points in sl(2,R)+ . In particular, Umb is additively closed, so, in view ofthe assertion in 6.7 we only have to consider the case where both H and −Hbelong to Umb. But the only semialgebra in sl(2,R)+ with inner points andcontaining R ·H is sl(2,R)+ itself. This finishes the proof.

6.10. Corollary. Let S be a Lie subsemigroup of Sl(2,R)+ whose Lie wedgeW is pointed. Then the following assertions hold:

(i) If S admits a normalized alien pair then dimW ≥ 2 and Q ∈W .

(ii) If {P,Q} ∩W = ∅ then S is a perfect semigroup.

Proof. If dimS = 3 then we know from Proposition 10.11 of [4] that Wcontains all nilpotent elements of Umb(S), so that assertion (i) follows fromCorollary 6.9. The case dimS = 2 is covered by 6.8 (note that in this caseW = Umb(S)), obviously the cases dimS = 0, 1 cannot occur.

Assertion (ii) follows from (i) and its application to the conjugate jSj .

6.11. Example. The condition {P,Q} ∩W = ∅ is by no means necessaryfor the perfectness of the Lie semigroup with Lie wedge W . In fact, if W1 isthe Lie wedge of a perfect Lie subsemigroup of Sl(2,R)+ then by 6.6 the Liesemigroup whose Lie wedge is generated by W1 and the set {P,Q} is perfect. Inparticular, the Lie semigroup with Lie wedge R+

0 P+R+0 Q (this is the semigroup

of all matrices in Sl(2,R)+ whose diagonal entries are ≥ 1) is perfect. (see also[21], 4.1(ii) and [4], 6.6).

6.12. For the next two propositions we recall some definitions. Let W be awedge in a finite dimensional vector space. A closed face f of W is said tobe exposed if there exists a support hyperplane p of W such that p ∩W = f .A point x is said to be a C1 -point of W if there exists exactly one supporthyperplane of W at x in the vector space W −W (cf. [11], p. 20f).

6.13. Proposition. Let S be a Lie semigroup in Sl(2,R)+ and assume thatthe Lie wedge W of S is pointed and polyhedral. Then

(i) S admits a normalized alien pair if and only if R+0 · Q lies in W but is

not an exposed closed face of W .

(ii) S is perfect if and only if each of the points P , Q either does not belongto W or generates an exposed face of W .


Proof. (i) Suppose first that S admits a normalized alien pair. Then by6.10(i), Q ∈ W . If R+

0 Q is an exposed closed face of W then there are pointsX1, X2, . . . , Xn ∈ sl(2,R)+ such that Q /∈ W1 = R+

0 X1+R+0 X2+· · ·R+

0 Xn and

W = W1 + R+0 Q . Since Q /∈ W1 we know that the Lie semigroup 〈exp(W1)〉

does not contain normalized alien pairs (6.10(i)). Now we only have to apply6.6 to the semigroup 〈exp(W1)〉 and M = exp(R+

0 Q).

Conversely, if R+0 Q lies in W but is not an exposed closed face of W

then Q must live in the algebraic interior of a face f of W . This face must havethe form R+

0 (−H + γQ) +R+0 (H + γ′Q), with γ, γ′ positive. In this case exp f

is a closed semigroup and admits a normalized alien pair.

Assertion (ii) is an immediate consequence of (i).

6.14. Proposition. Let S be a Lie semigroup in Sl(2,R)+ with pointed Liewedge W and assume that S admits a normalized alien pair. Then dimW ≥ 2and exactly one of the following assertions holds:

(i) If dimW = 2 then Q lies in the algebraic interior of W .

(ii) If dimW = 3 then Q is a C1 -point of W .

Proof. By 6.10(i) we know that Q ∈ W . Since W ⊆ sl(2,R)+ the planeRH + RQ supports W . If dimW < 3 then the assertion follows from 6.13(i).Suppose now that dimW = 3 and that Q is not a C1 -point of W . Thenthere exists a support plane p of W at Q different from RH + RQ . Let Vbe the half space bounded by p which contains W . Then Q generates anexposed face of the polyhedral wedge W1 = V ∩ sl(2,R)+ , so the semigroupS1 = 〈exp(W1)〉 does not admit normalized alien pairs, by 6.13(i). Since S is aclosed subsemigroup of S1 this implies that S does not admit normalized alienpairs.

6.15. Corollary. Let S be a Lie semigroup in Sl(2,R) whose Lie wedge Wis three dimensional and assume that ∅ �= Al(S) �= S . Then W contains anilpotent C1 -point.

6.16. Problem. If, conversely, W contains a nilpotent C1 -point, does itfollow that Al(S) �= ∅? (By 6.13 we know that Al(S) �= ∅ if the algebraicinterior of a two dimensional face of W contains a nilpotent element.)

6.17. Example. The above Corollary shows, in particular, that if a Liesubsemigroup S of Sl(2,R)+ with H(S) = {1} contains an alien elementthen it either contains expP or expQ . If we only assume that S is a closedsubsemigroup of Sl(2,R)+ with Al(S) �= ∅ then S need not contain unipotentelements at all. In fact, if x is a matrix in Sl(2,R)+ such that both diagonalentries are > 1 then the closure I of the ideal Sl(2,R)+xSl(2,R)+ contains alienelements (it is an ideal of the semigroup Sl(2,R)+ = Al(Sl(2,R)+), cf. 4.5(ii))but no unipotent matrices. In fact, I is properly contained in the interior of


Sl(2,R)+ . To see this, pick s, s′ ∈ Sl(2,R)+ and let µ be the minimum of thediagonal entries of x , by assumption µ > 1. Then, entry by entry, sxs ′ ≥ µ ·ss ′ .The product of the diagonal entries of the matrix ss′ is ≥ 1 (as all entries arenonnegative this follows from det(ss′) = 1) so the product of the diagonalentries in sxs ′ is ≥ µ2 > 1. Thus for every matrix in I the product of diagonalentries is ≥ µ2 > 1. Since the boundary of Sl(2,R)+ in Sl(2,R) is formedprecisely by those matrices where the product of the diagonal entries is = 1this implies the assertion.

7. Aliens in exponential subsemigroups of Sl(2,R)

7.1. In this section we explicitly describe the set Al(S) for exponential sub-semigroups S of Sl(2,R)+ .

7.2. The conjugacy classes of exponential subsemigroups of Sl(2,R)+

Every exponential subsemigroup S of Sl(2,R)+ with dimS ≥ 2 is conjugate toexactly one of the following semigroups or its transpose (the cases dimS = 0, 1are trivial).

1. dimS = 2:

(i) the half space semigroup exp(RH + R+0 Q);

(ii) the quadrant semigroup exp(R+0 H + R+

0 Q) or exp(−R+0 H + R+

0 Q);

(iii) the ‘Q-free’ version exp(R+0 H+R+

0 (H+2Q)) or exp(−R+0 H+R+

0 (−H+2Q);

(iv) the semigroup S1∗ =

{(a 0

c a−1

)∈ Sl(2,R)+ | a+ c ≥ a−1

},

with Lie wedge W = R+0 H + R+

0 (−H + 2Q) (which contains Q);

2. dimS = 3:

(v) the whole semigroup Sl(2,R)+ ;

(vi) the semigroup S1 =

{(a b

c d

)∈ Sl(2,R)+ | a+ c ≥ b+ d

},

with Lie wedge R+0 H + R+

0 (H + 2P ) + R+0 (−H + 2Q);

(vii) S1λ =

{(a b

c d

)∈ Sl(2,R)+ | a+ c ≥ b+ d and a+ 1

λb ≥ λc+ d

},

for some λ ∈ R+ .


(Cf. 7.14 of [4] for the case dimS = 3. For dimS = 2 the assertion can bededuced either by an easy direct computation or by applying the classificationfor dimS = 3.) We observe that S1∗ is the intersection of S1 with the Borelgroup exp(RH + RQ), in fact, S1∗ is a face of S1 .

7.3. Nonperfect exponential subsemigroups of Sl(2,R)+

Let S be a nonperfect exponential subsemigroup S of Sl(2,R)+ . If dimH(S) =1 then H(S) is noncompact, hence Al(S) = S . Assume now that the Liewedge W of S is pointed. Then dimS ≥ 2 and by 6.13(ii) either P or Q iscontained in the algebraic interior of a face of W . Inspecting the above listthis means that S must be conjugate to one of the semigroups S1∗ , S1 , or itstranspose.

In order to apply it more than once we first record a useful remark ondecompositions.

7.4. Proposition. In Sl(2,R)+ no decomposition of the form

(a b

c c

)=

(a′ b′

c′ d′

)(a′′ b′′

c′′ d′′

),

with c′ ≥ d′ and c′′ ≥ d′′ , is possible.

Proof. Dividing both sides of the above equation (from the left) by the

matrix(a′ b′

c′ d′

)we see that c′′ = c(d′ − c′). Since c′ ≥ d′ and c′′ ≥ 0 it follows

that c′ = d′ and therefore c′′ = 0 But c′′ ≥ d′′ and d′′ ≥ 0, so it follows thatc′′ = d′′ = 0, a contradiction.

7.5. Aliens in S1∗ . For the semigroup

S1∗ =

{(a 0

c a−1

)∈ Sl(2,R)+

∣∣∣∣∣ a+ c ≥ 1

a

}

the following assertions hold:

(i) Al(S1∗) =

{(a 0

c 1a

)∈ Sl(2,R)+

∣∣∣∣∣c ≥ 1a

}.

(ii) Al(S1∗) = exp(RH) exp([1,∞[Q) = exp(RH) exp(Q) exp(R+0 H) ; this set

is a half-space of the Borel group exp(RH+RQ) , bounded by the left cosetexp(RH) exp(Q) .


(iii) Al(Al(S1∗)) = Al(S1∗) and

PAl(Al(S1∗)) = exp(RH) exp(]1,∞[Q) =

{(a 0

c 1a

)∈ Sl(2,R)+

∣∣∣∣c > 1

a

}.

Proof. (i) Let s =(a 0c 1/a

)∈ Sl(2,R)+ with c ≥ 1

a . For every n ∈ N put

(∗) xn =

(n 01n

1n

), yn =

(an 0

cn na

), so that xnyn =

(a 0

an2 + c 1

a

)→ s.

It is clear that xn ∈ S1∗ . Since c ≥ 1a , the matrix yn also lies in S1∗ . Thus

s ∈ Al(S1∗).Conversely, suppose that s =

(a 0c 1/a

)∈ Al(S1∗) and consider a normal-

ized alien pair 〈xnyn〉 of S1∗ with s = limxnyn . We write

xn =

(an 0

cn1an

), yn =

(a′n 0

c′n1a′n

).

The inequality c′n ≥ ( 1a′n− a′n) implies

cna′n +

c′nan≥(

1

ana′n− a′nan

).

Passing to limits (taking into account that lim 1an

= lim a′n = 0) yields that

c ≥ 1a .

(ii) The first equality follows from (i) and the fact that the element(a 0

0 1a

)(1 0

c 1

)=

(a 0ca

1a

)

lies in Al(S1∗) if and only if a ∈ R+, c ≥ 1. Similarly, the remaining equalityAl(S1∗) = exp(RH) exp(Q) exp(R+

0 H) follows from the decomposition, valid forany a, c > 0,(

a 0

c 1a

)=

(√a/c 0

0√c/a

)(1 0

1 1

)(√ac 0

0 1/√ac

).

(iii) The inclusion Al(Al(S1∗)) ⊆ Al(S1∗) holds by definition. For thereverse inclusion choose an arbitrary element

(a 0c 1

a

)∈ Al(S1∗). By (i) the el-

ements xn and yn of (∗) belong to Al(S1∗), and limxnyn =(a 0c 1

a

). Thus

Al(Al(S1∗)) = Al(S1∗). The characterization of PAl(S1∗) follows from Propo-sition 7.4 and the calculations in (∗) if we replace c by c − a/n2 . (Note thatc− a/n2 > 1/a for all sufficiently large n .)


7.6. Aliens in S1 . For the semigroup

S1 =

{(a b

c d

)∈ Sl(2,R)+

∣∣∣∣∣ a+ c ≥ b+ d

}

the following assertions hold:

(i) Al(S1) =

{(a b

c d

)∈ S1

∣∣∣∣∣c ≥ d

}.

(ii) Al(S1∗) = Al(S1) ∩ S1∗ .

(iii) Al(S1) is the ideal generated by Al(S1∗) , i.e., Al(S1) = S1 Al(S1∗)S1 .In fact, we even have Al(S1) = Al(S1∗)S1 .

(iv) Al(Al(S1)) = Al(S1).

(v) PAl(Al(S1)) =

{(a b

c d

)∈ S1

∣∣∣∣∣c > d

}.

Proof. We first recall from [4], 6.11, that the matrix(a bc d

)∈ Sl(2,R)+

belongs to S1 if and only if (a− b)(a+ c) ≥ 1. Indeed, the inequalities

a+ c ≥ b+ d and (a− b)(a+ c) ≥ 1

are equivalent, since multiplying the first inequality with a > 0 yields

a(a+ c) ≥ a(b+ d) = ab+ bc+ 1 = b(a+ c) + 1,

and this is equivalent to (a− b)(a+ c) ≥ 1. In particular, for every(a bc d

)∈ S1

we have a > b .

(i) Pick an arbitrary s =(a bc d

)∈ S1 such that c ≥ d . For n ∈ N put

(∗∗) xn =

(n 01n

1n

), yn =

(an

bn

cn dn

).

Then clearly xn ∈ S1 ; since c ≥ d the matrix yn belongs to S1 , too. Fromlimxnyn = s we conclude that s ∈ Al(S1).

Conversely, choose s =(a bc d

)∈ Al(S1) and let 〈xn, yn〉 be a normalized

alien pair such that limxnyn = s . Write

xn =

(an bn

cn dn

), yn =


).


Using the inequalities a′n > b′n and c′n−d′n ≥ b′n−a′n we can write the followingrelations

cna′n+dnc

′n ≥ cnb

′n+dnc

′n = cnb

′n+dnd

′n+dn(c

′n−d′n) ≥ cnb

′n+dnd

′n+dn(b

′n−a′n).

Thus

cna′n + dnc

′n ≥ cnb

′n + dnd

′n + andn

b′n − a′nan

.

Passing to limits (taking into account that lim andn exists and that lim 1an

=lim b′n = lim a′n = 0), we get that c ≥ d .

(ii) We know by (i) that

Al(S1) ∩ S1∗ =

{(a b

c d

)∈ S1∗

∣∣∣∣∣c ≥ d

}.

Applying 7.5(i) the equality Al(S1∗) = Al(S1) ∩ S1∗ follows.

(iii) The inclusion S1 Al(S1∗)S1 ⊆ Al(S1) holds since Al(S1) is an idealof S1 containing Al(S1∗). Now pick an arbitrary element s =

(a bc d

)∈ Al(S1).

Then c ≥ d , by (i), and s decomposes within S1 as

(a b

c d

)=

(a′ 0

c′ 1a′

)(a′′ a′′ − 1

a′′

0 1a′′

)

where a′ = a√a−b√a

, c′ = c√a−b√a

, a′′ =√a√

a−b . The inequality c ≥ d implies

c′ ≥ 1a′ . Thus, by 7.5(i), s ∈ Al(S1∗)S1 and hence Al(S1) = Al(S1∗)S1 .

(iv) We have only to check the inclusion Al(S1) ⊆ Al(Al(S1)). Choosean arbitrary s =

(a bc d

)∈ Al(S1). By (i) the elements xn and yn of (∗∗) belong

to Al(S1), and limxnyn = s . This finishes the proof of (iv).

(v) Choose an element s =(a bc d

)∈ Al(S1) with c = d . Then by 7.4

the set LD(s) of left divisors of s in Al(S1) consists only of the element s , sos /∈ PAl(Al(S1)).

If s =(a bc d

)∈ Al(S1) with c > d then we have the following decomposi-

tion in Al(S1) for all sufficiently large positive reals n :

(a b

c d

)=

(n(a− b) b

n

n(c− d) dn

)(1n 0

n n

).

This decomposition shows that s ∈ PAl(Al(S1)) which finishes the proof.

7.7. Example. It is not generally true that for Lie subsemigroups S, T ofSl(2,R)+ the inclusion S ⊆ T implies Al(S) = Al(T ) ∩ S . To see this, let g =exp(H) and put S = S1∗ , T = gSg−1 . Then S ⊂ T and Al(T ) = gAl(S)g−1 .


However the matrix s =

(1 0

e−2 1

)lies in

Al(T ) ∩ S1∗ =

{(a 0

e−2c a−1

)∣∣∣∣∣c ≥ a−1, a+ e−2c ≥ a−1

}

but not in Al(S) = Al(S1∗) =

{(a 0

c 1a

)∈ Sl(2,R)+

∣∣∣∣∣c ≥ 1

a

}.

7.8. Remark. The assertions 7.5(ii) and 7.6(iii) show that the nonperfectsemigroups S1∗ and S1 are of the type described in 2.4(iv) (with G1 =exp(RH)). We do not yet know whether this phenomenon is typical for non-perfect Lie semigroups.

8. Aliens and the Bohr compactification

In this section we compute the Bohr compactification of the semigroups S1∗ andS1 , which are our prototypes for all exponential and nonperfect subsemigroupsof Sl(2,R). We start with some prerequisites from the theory of compactsemigroups.

8.1. Minimal ideals in compact semigroups

The following assertions about minimal ideals in compact semigroups are stan-dard facts (cf., e.g., [5], p. 22–26).

(i) Every compact topological semigroup S contains a unique minimal idealM(S) which is closed.

(ii) An idempotent e of S lies in the minimal ideal of S if and only if eSe isa group.

(iii) The following assertions are equivalent (they are always satisfied if S isabelian):

(a) M(S) contains only one idempotent.

(b) M(S) is a group.

(c) M(S) contains a central idempotent.

8.2. Notation. (i) If (K, i: S → K) is a compactification of the topologicalsemigroup S then, to simplify notation, we use a period “.” to denote the

natural actions of S on K , thus s.kdef= i(s)k and k.s

def= ki(s), whenever

s ∈ S , k ∈ K .

(ii) The Bohr compactification of a (topologized) semigroup S is denotedwith (Sb, ibS), or (Sb, iS), or S

b for short.


8.3. The join of two compactifications

(i) Recall that if (K1, i1) and (K2, i2) are two (topological semigroup) compact-ifications of a topological semigroup S then we can form the new compactifica-tion (K, i), where K = (i1 × i2)(S) and i: S → K denotes the corestriction ofi1 × i2 to K . The compactification (K, i) is the join of the compactifications(K1, i1) and (K2, i2) with respect to the natural order of compactifications.

(ii) If the two canonical quotient maps Sb → K1 and Sb → K2 , definedaccording to the universal property of the Bohr compactification, separate thepoints of Sb then the Bohr compactification and the join (K, i) are equivalent.

8.4. The universal topological group compactification

(i) Recall that a compactification (Sg, ig) of a (topologized) semigroup S iscalled the universal topological group compactification of S if Sg is a compacttopological group and every compactification whose compactification space is atopological group can be factorized over (Sg, ig).

(ii) The group Sg can be obtained by forming the quotient Sb/ ∼=, where∼= is the smallest closed congruence on Sb such that the minimal ideal Min Sb/ ∼= is a group. (In fact, ∼= is the smallest closed congruence suchthat all minimal idempotents are equivalent.) Then we put Sg = M anddefine ig = λm◦κ◦iS to be the composition of iS with the quotient morphismκ: Sb → Sb/ ∼= and the homomorphism λm: Sb/ ∼=→ M , x �→ mx , where mdenotes the identity of M .

(iii) It follows immediately from (ii) that if M(Sb) is a [left] [right] groupthen the universal topological group compactification of S is equivalent to thecompactification (H(m), ϕm◦iS), where m is an idempotent in M(Sb) andϕm: Sb → H(m), [x �→ mx ] [x �→ xm ]. (Recall that in a [left] [right] group allidempotents are [right] [left] identities.)

(iv) Alternatively, we can construct the universal topological group com-pactification of S by first forming the free topological group (G(S), γ) overS and then passing to the Bohr compactification G(S)b of G(S). (Warning:a priori G(S) need not be Hausdorff!) Then the universal topological groupcompactification of S is equivalent with the pair (G(S)b, iG(S)◦γ).

Proof of (iv): Note first that the closure of iG(S)◦γ(S) is a compacttopological group and that γ(S) generates G(S) as a group. Thus there is amorphism ϕ: Sg → G(S)b with ϕ◦ig = iG(S)◦γ .

By the definition of G(S) we can extend the morphism ig: S → Sg toa continuous homomorphism G(S) → Sg . Thus, by the universality of theBohr compactification, there is a canonical morphism G(S)b → Sg which is theinverse of ϕ . This proves the assertion.

(v) Let S be a connected subsemigroup with dense interior of a simplyconnected Lie group G . Then the universal topological group compactificationof S is equivalent with the compactification (Gb, iG◦i), where i is the embeddingmorphism of S into G .


(In view of (iv) we only have to observe that the pair (G(S), γ) is iso-morphic with (G, i), by Proposition VII.3.28 of [11].)

8.5. The Bohr compactification of R+0 . (Cf. [5] 5.18) The following

assertions hold for the Bohr compactification of S = R+0 :

(i) Sb = iS(S) ∪M(Sb) and M(Sb) is a group isomorphic with Rb .

(ii) If 〈rn〉 is a net in R+0 with lim rn =∞ then every cluster point of iS(rn)

lies in the minimal ideal M(Sb) .

(iii) The Bohr compactification of R+0 is the join of its one point compactifi-

cation and its universal topological group compactification.

8.6. Theorem. For S = S1∗ the Bohr compactification Sb has the followingproperties:

(i) The element m = iS(exp(Q)) is a central idempotent in Sb and mSb isa group. Thus mSb =M(Sb) .

(ii) For any x, y ∈ S we have m.x = m.y if and only if xy−1 ∈ exp(RQ) .

(iii) If 〈sn〉 is a net in S with lim sn = ∞ then each cluster point of the netiS(sn) belongs to M(Sb) .

(iv) M(Sb) = iS(Al(S)) .

(v) Write G for the Borel group exp(RH + RQ) and N for its normalsubgroup exp(RQ) . Let ϕ be the homomorphism ϕ:S → R which isthe restriction to S of the morphism G → G/N ∼= R . Then ϕ(S) =R and the compactification (Rb, iR◦ϕ) is the universal topological groupcompactification of S .

(vi) Sb is the join of the large ideal compactification and the universal topo-logical group compactification of S .

Proof. For the sake of notational convenience let us put A = −H + 2Q .For later use, in different contexts, we first record the following formulas, whichhold for any α, β, t ∈ R , X = αH + βQ :

(a) exp(X) exp(tQ) = exp(e−2αtQ) exp(X),

(b) exp(tA) exp(tH ) = exp((e2t − 1)Q),

(c) exp(tH) exp(tA) = exp((1− e−2t)Q).

Let m be the minimal idempotent of the compact abelian semigroup

iS(exp(R+0 Q)), we shall see later that m = iS(exp(Q)).

Step 1: m is central. Choose a net 〈tn〉 in R+ such that 〈iR

+0(tn))〉

converges to the minimal idempotent in (R+0 )

b . Clearly, tn → ∞ in R . Since


multiplication with a positive scalar is an isomorphism of the additive semigroupR+ and because of the universal property of the Bohr compactification we seethat lim iS(exp(tnQ)) = lim iS(exp(tne

−2αQ)) = m , for arbitrary α ∈ R . Inequation (a) let X be a vector lying in the Lie wedge of S and put t = tn . Then,applying iS on both sides and taking limits we get exp(X).m = m. exp(X).Thus S.m = m.S and, by taking closures, it follows that Sbm = mSb , so m iscentral in Sb .

Step 2: mS b is a group. By equations (b) and (c) above, for anyt > 0 the products exp(tH ) exp(tA) and exp(tA) exp(tH ) lie in exp(R+Q),thus m. exp(tH ) and m. exp(tA) lie in the maximal subgroup H(m) of m inSb . Since exp(R+

0 A) and exp(R+0 H) generate S this means that mS b = m.S

is a group. By Step 1, 8.1(ii) and 8.1(iii) this also implies that mS b =M(Sb).

Step 3: We show (ii) and (v). Assertion (v) follows from 8.4(v), sinceN = i−1

G (1). By Step 2 we know that M(Sb) is a group, so we concludethat the universal topological group compactification of S is equivalent with(M(Sb), λm◦iS), where λm: S → M(Sb), x �→ m.x . This means that m.x =m.y if and only if xy−1 ∈ N (recall that the map iR is injective).

Step 4: Let eA be the minimal idempotent in iS(exp(R+0 A)) and eH the

minimal idempotent in iS(exp(R+0 H)). We show that eA = eH = m . Let tn

be as in Step 1 above. Then formula (a) with α = −tn , β = 2tn and t replacedwith e−2tnt yields, for every t > 0,

eA = lim iS(exp(tnA)). exp(te−2tnQ) = lim exp(tQ).iS(exp(tnA)) = exp(tQ).eA

and we conclude eA = meA . Since m is central and lies in the minimal idealthis means that m = eA . Similarly, we feed formula (a) with α = tn , β = 0,t = t , and find

eH . exp(tQ)=lim iS(exp(tnH)). exp(tQ)=lim exp(e−2tntQ).iS(exp(tnH))=eH ,

which implies eH = eHm , and thus m = eH , since m is minimal.

Step 5: We show (iii). By the properties of the Bohr compactificationof R+

0 , as listed in 8.5, we also conclude from Step 4 that for any net 〈un〉in R+ with limun = ∞ all cluster points of the nets 〈iS(exp(unA))〉 and〈iS(exp(unH))〉 belong to mSb . Since S = exp(R+

0 A) exp(R+0 H) this estab-

lishes (iii). Assertion (iii) also implies (iv).

Step 6: iS(exp(Q)) = m , completing the proof of (i). This identity isseen from equation (c) with t = tn , after applying iS and passing to limits.

The final Step: Proof of (vi). By (iv) we know that Sb = iS(S\Al(S)) ∪M(Sb). Since the large ideal compactification separates the points of S\Al(S)and since (M(Sb), λm · iS) is equivalent with the universal topological groupcompactification of S , the above observation 8.3(ii) implies assertion (vi).

8.7. Notation. By definition the Lie wedge W 1 of S1 is the pointed cone

spanned by the three vectors H , Adef= −H+2Q and B

def= H+2P . To simplify


notation, for X,Y ∈ {A,B,H} we henceforth write

WXY = R+0 X + R+

0 Y and SXY = exp(WXY ).

Note that the sets SXY are exponential Lie semigroups and that W 1 is boundedby the three semialgebras WAB , WAH , and WBH . Note also that SAH = S1∗ .

8.8. The affine triangle

The semigroup Mn(2,R) of all 2×2-matrices over R contains the subsemigroup

∇ def=

{s =

(x y

0 1

)∈Mn(2,R) | x+ y ≤ 1, x > 0, y ≥ 0

},

whose closure ∇ is called the affine triangle, or the triangle for short (cf. [5],p. 11).

The following properties of ∇ are easily checked:

(i) For any λ ∈ [0, 1] the elements s ∈ ∇ with y = λ(1 − x) form a com-pact subsemigroup Iλ of ∇ , which is isomorphic with the multiplicativesemigroup [0, 1].

(ii) The minimal ideal M(∇) of ∇ is formed by the matrices s ∈ ∇ withx = 0, each such matrix is a left zero element, i.e., st = s for alls ∈M(∇), t ∈ ∇ . Moreover, M(∇) = I0

(0 10 1

).

(iii) If ϕ: ∇ → S is a surjective semigroup morphism mapping the minimalidempotents

(0 00 1

)and

(0 10 1

)onto the same point then

M(S)=ϕ(M(∇))=ϕ(I0)ϕ((

0 1

0 1

))=ϕ(I0)ϕ

((0 0

0 1

))=

{ϕ

((0 0

0 1

))}

is singleton.

(iv) The open dense subsemigroup ∇ of ∇ is an exponential Lie subsemigroupof Gl(2,R), its Lie wedge W∇ is spanned by the matrices X1 = −

(1 00 0

)and X2 = X1 + P .

8.9. Mappings into the affine triangle

(i) The linear map RA + RB → RX1 + RX2 which sends A to 2X1 and Bto 2X2 , maps [A,B] = 2(A − B) onto [2X1, 2X2] = 4(X1 − X2), hence isan isomorphism of Lie algebras mapping WAB onto W∇ . On the level of Liegroups this morphism induces an isomorphism SAB → ∇ .

(ii) Similarly, the linear map RH + RB → RX1 + RX2 with H �→ 2X1

and B �→ 2X2 is an anti-isomorphism of Lie algebras giving rise to an anti-isomorphism SBH → ∇ .


8.10. Compactifications of ∇

The proofs of the following assertions are left to the reader.

(i) The Bohr compactification (∇b, i∇) of ∇ is the join of the affinetriangle compactification and the universal topological group compactificationof ∇ . (It is known that ∇b = i∇(∇)∪M(∇b) and that M(∇b) is a left group,cf., e.g., 2.8.1 of [3]. Thus this assertion follows from 8.4(iii).)

(ii) The universal topological group compactification of ∇ is obtained

by combining the homomorphism ϕ: ∇ → R ,

(x y

0 1

)→ x , with the Bohr

compactification map iR (cf. 8.4(v)).

(iii) Note that, in particular, i∇ is an embedding and that M(∇b) is theremainder ∇b\i∇(∇) of the Bohr compactification.

(iv) If ϕ: ∇ → K is a compactification of ∇ such that the compact

abelian semigroups ϕ(exp(R+0 X1)) and ϕ(exp(R+

0 X2)) have the same minimalidempotent then M(K) is a group. (Since we may assume, without losinggenerality, that the maximal subgroups in the minimal ideal of K are singleton,this follows by the universality of the Bohr compactification, property 8.8(iii),and the construction of ∇b in (i).)

In view of 8.9 and 8.10 the Bohr compactifications of SAB and SBH canbe considered as known.

8.11. Theorem. Let S be a three dimensional exponential subsemigroup ofSl(2,R) such that the Lie wedge of S contains exactly one nonzero nilpotentelement. Then the large ideal compactification and the Bohr compactification ofS are equivalent.

Proof. Let (Sb, iS) be the Bohr compactification of S . We may assume,without loss of generality, that S = S1 . For X ∈ {A,B,H} denote with

eX the minimal idempotent of the compact abelian semigroup iS(exp(R+0 X)).

From 8.6(i) we conclude that eA = eH . Since the idempotents in the minimalideal of iS(SAB ) form a left zero semigroup we conclude eA = eAeB .

Since SBH is anti-isomorphic with SAB the idempotents in the minimalideal of iS(SBH ) form a right zero semigroup, so we conclude eB = eHeB . ButeH = eA , so eB = eAeB = eA . By 8.10(iv) we conclude that the minimalideals of the semigroups iS(SAB ), iS(SAH ), iS(SBH ) are subgroups of a singlemaximal subgroup H(e), where e = eA = eB = eH . Since S is generated bySAH , SAB and SBH this implies that H(e) = M(Sb). Since the morphismS → H(e), x �→ eiS(x) can be extended to a continuous homomorphism of

Sl(2,R) we conclude that H(e) must be singleton.

We recall (see, for example, 6.8 and 6.11 of [4]) that S = SAHSBHSAB ,hence

(∗) Sb = iS(SAH )iS(SBH )iS(SAB ).


This means that for every net 〈sn〉 in S with lim sn = ∞ we must havelim iS(sn) = e . It follows that iS(Al(S)) = {e} , so the natural morphismSb → S� separates the points of Sb and the assertion follows.

Acknowledgments

We want to thank especially Karl Heinrich Hofmann with whom we hadmany fruitful conversations about the subject. The Deutsche Forschungsge-meinschaft (DFG) provided financial support to the first author during her stayat Darmstadt and at Vienna while working on this project, and the Institute ofMathematics and Applied Statistics at the University of Bodenkultur at Viennaprovided the necessary infrastructure.

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Babes-Bolyai UniversityFaculty of Mathematics andComputer Science

Str. M. Kogalniceanu 1RO-3400 Cluj-Napoca, [email protected]

Institut fur Mathematik undAngewandte Statistik

Universitat fur BodenkulturPeter Jordanstr. 82A-1190 Wien, [email protected]

Institut fur Mathematik undAngewandte Statistik

Universitat fur BodenkulturPeter Jordanstr. 82A-1190 Wien, [email protected]

Received November 11, 2001and in final form March 19, 2002Online publication May 3, 2002

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