Open subgroups of locally compact Kac-Moody groups
aa r X i v : . [ m a t h . G R ] A ug OPEN SUBGROUPS OF LOCALLY COMPACT KAC–MOODYGROUPS
PIERRE-EMMANUEL
CAPRACE ∗ AND TIMOTH´EE
MARQUIS ‡ Abstract.
Let G be a complete Kac-Moody group over a finite field. It isknown that G possesses a BN-pair structure, all of whose parabolic subgroupsare open in G . We show that, conversely, every open subgroup of G is containedwith finite index in some parabolic subgroup; moreover there are only finitelymany such parabolic subgroups. The proof uses some new results on parabolicclosures in Coxeter groups. In particular, we give conditions ensuring that theparabolic closure of the product of two elements in a Coxeter group containsthe respective parabolic closures of those elements. Introduction
This paper is devoted to the study of open subgroups of complete Kac–Moodygroups over finite fields. The interest in the structure of those groups is motivatedby the fact that they constitute a prominent family of locally compact groupswhich are simultaneously topologically simple and non-linear over any field (see[R´em04]) and [CR09]). They show some resemblance with the simple linear lo-cally compact groups arising from semi-simple algebraic groups over local fieldsof positive characteristic.The first question on open subgroups of a given locally compact group G onemight ask is: How many such subgroups are there? Let us introduce some ter-minology providing possible answers to this question. We say that G has fewopen subgroups if every proper open subgroup of G is compact. We say that G is Noetherian if G satisfies an ascending chain condition on open subgroups.Equivalently G is Noetherian if and only if every open subgroup of G is compactlygenerated (see Lemma 3.22 below). Clearly, if G has few open subgroups, thenit is Noetherian. Basic examples of locally compact groups that are Noetherian— and in fact, even have few open subgroups — are connected groups and com-pact groups. Noetherianity can thus be viewed as a finiteness condition whichgeneralizes simultaneously the notion of connectedness and of compactness. It ishighlighted in [CM11], where it is notably shown that a Noetherian group admitsa subnormal series with every subquotient compact, or abelian, or simple. An ex-ample of a non-Noetherian group is given by the additive group Q p of the p -adics.Other examples, including simple ones, can be constructed as groups acting ontrees.According to a theorem of G. Prasad [Pra82] (which he attributes to Tits),simple locally compact groups arising from algebraic groups over local fields havefew open subgroups. Locally compact Kac–Moody groups are however known tohave a broader variety of open subgroups in general. Indeed, Kac–Moody groups Date : August 2011. ∗ F.R.S.-FNRS Research Associate, supported in part by FNRS grant F.4520.11. ‡ F.R.S.-FNRS Research Fellow. are equipped with a BN -pair all of whose parabolic subgroups are open. In par-ticular, if the Dynkin diagram of a Kac–Moody group admits proper subdiagramsthat are not of spherical type, then the corresponding Kac–Moody groups haveproper open subgroups that are not compact.Our main result is that parabolic subgroups in Kac–Moody groups are essen-tially the only source of open subgroups. Theorem A.
Every open subgroup of a complete Kac-Moody group G over a finitefield has finite index in some parabolic subgroup.Moreover, given an open subgroup O , there are only finitely many distinct par-abolic subgroups of G containing O as a finite index subgroup. A more precise statement of this theorem will be given later, see Theorem 3.3.As a consequence, we deduce the following.
Corollary B.
Complete Kac–Moody groups over finite fields are Noetherian.
In fact, Theorem A allows us to characterize those locally compact Kac–Moodygroups having few open subgroups, as follows.
Corollary C.
Let G be a complete Kac–Moody group of irreducible type over afinite field. Then G has few open subgroups if and only if the Weyl group of G isof affine type, or of compact hyperbolic type. Notice that the list of all compact hyperbolic types of Weyl groups is finite andcontains diagrams of rank at most 5 (see e.g.
Exercise V.4.15 on p. 133 in [Bou68]).The groups in Corollary C include in particular all complete Kac–Moody groupsof rank two.Another application of Theorem A is that it shows how the BN -pair structureis encoded in the topological group structure of a Kac–Moody group. Here is aprecise formulation of this. Corollary D.
Let G be a complete Kac–Moody group over a finite field and P < G be an open subgroup. If P is maximal in its commensurability class, then P is aparabolic subgroup of G . Our proof of Theorem A relies on some new results on parabolic closures inCoxeter groups, which we now proceed to describe. Let thus (
W, S ) be a Coxetersystem with S finite. Recall that any intersection of parabolic subgroups in W isitself a parabolic subgroup. Following D. Krammer [Kra09], it thus makes sense todefine the parabolic closure of a subset of W as the intersection of all parabolicsubgroups containing it. The parabolic closure of a set E ⊆ W is denoted byPc( E ). Theorem E.
Let w ∈ W be an element of infinite order and let λ be a translationaxis for w in the Davis complex. Assume that the parabolic closure Pc( w ) is ofirreducible type.Then there is a constant C such that for any two parallel walls m, m ′ transverseto λ , if d ( m, m ′ ) > C , then Pc( w ) = Pc( r m , r m ′ ) . In particular, we get the following.
Corollary F.
Any irreducible non-spherical parabolic subgroup of a Coxeter groupis the parabolic closure of a pair of reflections.
PEN SUBGROUPS OF KAC–MOODY GROUPS 3
Our main result on Coxeter groups concerns the parabolic closure of the productof two elements.
Theorem G.
There is a finite index normal subgroup W < W enjoying thefollowing property.For all g, h ∈ W , there exists a constant K = K ( g, h ) ∈ N such that for all m, n ∈ Z with min {| m | , | n | , | m/n | + | n/m |} ≥ K , we have Pc( g m h n ) ⊇ Pc( g ) ∪ Pc( h ) . The following corollary is an essential ingredient in the proof of Theorem A.
Corollary H.
Let H be a subgroup of W . Then there exists h ∈ H such that theparabolic closure of h has finite index in the parabolic closure of H . Walls and parabolic closures in Coxeter groups
Throughout this section, we let (
W, S ) be a Coxeter system with W finitelygenerated (equivalently S is finite). Let Σ be the associated Coxeter complex,and let | Σ | denote its standard geometric realization. Also, let X be the Davisrealization of Σ. Thus X is a CAT(0) subcomplex of the barycentric subdivisionof | Σ | .Let Φ = Φ(Σ) denote the set of half-spaces of Σ. A half-space α ∈ Φ will alsobe called a root . Given a root α ∈ Φ, we write r α = r ∂α for the unique reflectionof W fixing the wall ∂α of α pointwise.We say that two walls m, m ′ of X are parallel if either they coincide or theyare disjoint. We say that the walls m, m ′ are perpendicular if they are distinctand if the reflections r m and r m ′ commute.Finally, for a subset J ⊆ S , we set J ⊥ := { s ∈ S \ J | sj = js ∀ j ∈ J } .In this paper, we call a subset J ⊆ S essential if each irreducible componentof J is non-spherical.2.1. The normalizer of a parabolic subgroup.Lemma 2.1.
Let L ⊆ S be essential. Then N W ( W L ) = W L × Z W ( W L ) and isagain parabolic. Moreover, Z W ( W L ) = W L ⊥ . Proof . See [Deo82, Proposition 5.5] and [Kra09, Chapter 3]. (cid:3)
Preliminaries on parabolic closures.
A subgroup of W of the form W J forsome J ⊂ S is called a standard parabolic subgroup . Any of its conjugates iscalled a parabolic subgroup of W . Since any intersection of parabolic subgroupsis itself a parabolic subgroup (see [Tit74]), it makes sense to define the parabolicclosure Pc( E ) of a subset E ⊂ W as the smallest parabolic subgroup of W containing R . For w ∈ W , we will also write Pc( w ) instead of Pc( { w } ). Lemma 2.2.
Let G be a reflection subgroup of W , namely a subgroup of W generated by a set T of reflections. We have the following: (i) There is a set of reflections R ⊂ G , each conjugate to some element of T ,such that ( G, R ) is a Coxeter system. (ii) If T has no nontrivial partition T = T ∪ T such that [ T , T ] = 1 , then ( G, R ) is irreducible. (iii) If ( G, R ) is irreducible (resp. spherical, affine of rank ≥ ), then so is Pc( G ) . P-E. CAPRACE AND T. MARQUIS (iv) If G ′ is a reflection subgroup of irreducible type which centralizes G and if G is of irreducible non-spherical type, then either Pc( G ∪ G ′ ) ∼ = Pc( G ) × Pc( G ′ ) or Pc( G ) = Pc( G ′ ) is of irreducible affine type. Proof . For (i) and (iii), see [Cap09, Lemma 2.1]. Assertion (ii) is easy to verify.For (iv), see [Cap09, Lemma 2.3]. (cid:3)
Lemma 2.3.
Let α ( α ( · · · ( α k be a nested sequence of half-spaces suchthat A = h r α i | i = 0 , . . . , k i is infinite dihedral. If k ≥ , then for any wall m which meets every ∂α i , either r m centralizes Pc( A ) , or h A ∪ { r m }i is a Euclideantriangle group. Proof . This follows from [Cap06, Lemma 11] together with Lemma 2.2(iv). (cid:3)
Parabolic closures and finite index subgroups.Lemma 2.4.
Let H < H be subgroups of W . If H is of finite index in H , then Pc( H ) is of finite index in Pc( H ) . Proof . For i = 1 ,
2, set P i := Pc( H i ). Since the kernel N of the action of H on thecoset space H /H is a finite index normal subgroup of H that is contained in H ,so that in particular Pc( N ) ⊆ Pc( H ), we may assume without loss of generalitythat H is normal in H . But then H normalizes P . Up to conjugating by anelement of W , we may also assume that P is standard, namely P = W I for some I ⊆ S . Finally, it is sufficient to prove the lemma when I is essential, whichwe assume henceforth. Lemma 2.1 then implies that P < W I × W I ⊥ . We thushave an action of H on the residue W I × W I ⊥ , and since H stabilizes W I andhas finite index in H , the induced action of H on W I ⊥ possesses finite orbits.By the Bruhat–Tits fixed point theorem (see for example [AB08, Th.11.23]), itfollows that H fixes a point in the Davis realization of W I ⊥ , that is, it stabilizesa spherical residue of W I ⊥ . This shows [ P : W I ] < ∞ . (cid:3) Parabolic closures and essential roots.
Our next goal is to present a de-scription of the parabolic closure Pc( w ) of an element w ∈ W , which is essentiallydue to D. Krammer [Kra09].Let w ∈ W . A root α ∈ Φ is called w -essential if either w n α ( α or w − n α ( α for some n >
0. A wall is called w -essential if it bounds a w -essential root. Wedenote by Ess( w )the set of w -essential walls. Clearly Ess( w ) is empty if w is of finite order. If w is of infinite order, then it acts on X as a hyperbolic isometry and thus possessessome translation axis. We say that a wall is transverse to such an axis if itintersects this axis in a single point. We recall that the intersection of a walland any geodesic segment which is not completely contained in that wall is eitherempty or consists of a single point (see [NV02, Lemma 3.4]). Given x, y ∈ X , wesay that a wall m separates x from y if the intersection [ x, y ] ∩ m consists of asingle point. Lemma 2.5.
Let w ∈ W be of infinite order and let λ be a translation axis for w in X . Then Ess( w ) coincides with those walls which are transverse to λ . PEN SUBGROUPS OF KAC–MOODY GROUPS 5
The proof requires a subsidiary fact. Recall that Selberg’s lemma ensures thatany finitely generated linear group over C admits a finite index torsion-free sub-group. This is thus the case for Coxeter groups. The following lemma providesimportant combinatorial properties of those torsion-free subgroups of Coxetergroups. Throughout the rest of this section, we let W < W be a torsion-freefinite index normal subgroup. Lemma 2.6.
For all w ∈ W and α ∈ Φ , either wα = α or w.∂α ∩ ∂α = ∅ . Proof . See Lemma 1 in [DJ99]. (cid:3)
Proof of Lemma 2.5.
It is clear that if α ∈ Φ is w -essential, then ∂α is tranverseto any w -axis. To see the converse, let n > w n ∈ W . Since λ isalso a w n -axis, we deduce from Lemma 2.6 that for all roots α such that ∂α istransverse to λ , we have either w n α ( α or α ( w n α . The result follows. (cid:3) We also set Pc ∞ ( w ) = h r α | α is a w -essential root i . Notice that every nontrivial element of W is hyperbolic. Moreover, in view ofLemma 2.6, we deduce that if w ∈ W , then a root α is w -essential if and only if wα ( α or w − α ( α . Lemma 2.7.
Let w ∈ W be of infinite order, let λ be a translation axis for w in X and let x ∈ λ .Then we have the following.(i) Pc ∞ ( w ) = h r α | ∂α is a wall transverse to λ i = h r α | ∂α is a wall transverse to λ and separates x from wx i . (ii) Pc ∞ ( w ) coincides with the essential component of Pc( w ) , i.e. the product ofits non-spherical components. In particular Pc( w ) = Pc ∞ ( w ) if and only if Pc( w ) is of essential type.(iii) If w ∈ W , then Pc( w ) = Pc ∞ ( w ) . Proof . The first equality in Assertion (i) follows from Lemma 2.5. To check thesecond, it suffices to remark that if ∂α is any wall transverse to λ , then thereexists a power w k of w such that w k ∂α separates x from wx .Assertion (ii) follows from Corollary 5.8.7 in [Kra09] (notice that what we call essential roots here are called odd roots in loc. cit. ). Assertion (iii) follows fromLemma 2.6 and Theorem 5.8.3 from [Kra09]. (cid:3) The Grid Lemma.
The following lemma is an unpublished observation dueto the first author and Piotr Przytycki.
Lemma 2.8 (Caprace–Przytycki) . There exists a constant N , depending only on ( W, S ) , such that the following property holds. Let α ( α ( . . . α k and β ( β ( · · · ( β l be two nested families of half-spaces of X such that min { k, l } > N .Set A = h r α i | i = 0 , . . . , k i , A ′ = h r α i | i = N, N + 1 , . . . , k − N i , B = h r β j | j =0 , . . . , l i and B ′ = h r β j | j = N, N + 1 , . . . , k − N i . If ∂α i meets ∂β j for all i, j ,then either of the following assertions holds: (i) The groups A and B are both infinite dihedral, their union generates aEuclidean triangle group and the parabolic closure Pc( A ∪ B ) coincideswith Pc( A ) and Pc( B ) and is of irreducible affine type. P-E. CAPRACE AND T. MARQUIS (ii)
The parabolic closures
Pc( A ) , Pc( A ′ ) , Pc( B ) and Pc( B ′ ) are all of irre-ducible type; furthermore we have Pc( A ′ ∪ B ) ∼ = Pc( A ′ ) × Pc( B ) and Pc( A ∪ B ′ ) ∼ = Pc( A ) × Pc( B ′ ) . We shall use the following related result.
Lemma 2.9.
There exists a constant L , depending only on ( W, S ) , such that thefollowing property holds. Let α ( α ( . . . α k be a nested sequence of half-spacesand m, m ′ be walls such that ∅ = m ∩ m ′ ⊂ ∂α , and that both m and m ′ meets ∂α i for each i . If k ≥ L , then h r m , r m ′ , r α i | i = 0 , . . . , k i is a Euclidean trianglegroup and h r α i | i = 0 , . . . , k i is infinite dihedral. Proof . See [Cap06, Th. 8]. (cid:3)
Proof of Lemma 2.8.
We let N = max { , L } where L is the constant appearingin Lemma 2.9.Assume first that for some i ∈ { , , . . . , k } and some j ∈ { N, N + 1 , . . . , l − N } ,the reflections r α i and r β j do not centralize one another. Let φ = r α i ( β j ); thus φ
6∈ {± α i , ± β j } . Let x ∈ ∂α ∩ ∂β j and x k ∈ ∂α k ∩ ∂β j . Then the geodesicsegment [ x , x k ] lies entirely in ∂β j and crosses ∂α i . Since ∂α i ∩ ∂β j is containedin ∂φ , it follows that [ x , x k ] meets ∂φ . This shows that the wall ∂φ separates x from x k .Let now p ∈ ∂α ∩ ∂β and p k ∈ ∂α k ∩ ∂β . Then the piecewise geodesic path[ x , p ] ∪ [ p , p k ] ∪ [ p k , x k ] is a continuous path joining x to x k . This path musttherefore cross ∂φ . Thus ∂φ meets either ∂α or ∂β or ∂α k . We now deal withthe case where ∂φ meets ∂α . The other two cases may be treated with analogousarguments; the straightforward adaption will be omitted here.Then ∂φ meets ∂α m for each m = 0 , , . . . , i . Therefore Lemma 2.9 may beapplied, thereby showing that A i = h r α m | m = 0 , . . . , i i is infinite dihedral andthat the subgroup T = h r α m , r β j | m = 0 , . . . , i i is a Euclidean triangle group.Furthermore Lemma 2.2(iii) shows that Pc( T ) is of irreducible affine type. SincePc( A i ) is infinite (because A i is infinite) and contained in Pc( T ) (because A i is contained in T ), it follows that Pc( A i ) = Pc( T ) since any proper parabolicsubgroup of Pc( T ) is finite. We set P := Pc( A i ) = Pc( T ).Let now n ∈ { , , . . . , l } with n = j . Then r β n does not centralize r β i ; inparticular it does not centralize T . On the other hand the wall ∂β n meets ∂α m for all m = 0 , . . . , i , which implies by Lemma 2.3 that h A i ∪ { r β n }i is a Euclideantriangle group. Therefore r β n ∈ P by Lemma 2.2(iii).We have already seen that P is of irreducible affine type. We have just shownthat B is contained in Pc( A i ) = P ; in particular this shows that B is infinitedihedral since the walls ∂β , . . . ∂β l are pairwise parallel. Moreover, the group h B ∪ { r α i }i must be a Euclidean triangle group since it is a subgroup of P . Inparticular we have Pc( B ) = P by Lemma 2.2(iii). Since every ∂α m meets every ∂β j , the same arguments as before now show that r α m ∈ Pc( B ) = P for all m = i + 1 , . . . , k . Finally we conclude that Pc( A ) = Pc( B ) = P in this case.Notice that, in view of the symmetry between the α ’s and the β ’s, the previousarguments yield the same conclusion if one assumed instead that for some i ∈{ N, N + 1 , . . . , k − N } and some j ∈ { , , . . . l } , the reflections r α i and r β j do notcentralize one another. PEN SUBGROUPS OF KAC–MOODY GROUPS 7
Assume now that for all i ∈ { , . . . , k } and all j ∈ { N, N + 1 , . . . , l − N } , thereflections r α i and r β j commute and that, furthermore, for all i ∈ { N, N +1 , . . . , k − N } and all j ∈ { , , . . . l } , the reflections r α i and r β j commute. By Lemma 2.2(ii)the parabolic closures Pc( A ), Pc( A ′ ) Pc( B ) and Pc( B ′ ) are of irreducible type.By assumption A ′ centralizes B . By Lemma 2.2(iv), either Pc( A ′ ) = Pc( B ) is ofaffine type or else Pc( A ′ ∪ B ) ∼ = Pc( A ′ ) × Pc( B ). In the former case, we may argueas before to conclude again that Pc( A ) = Pc( B ) is of affine type and we are incase (i) of the alternative. Otherwise, we have Pc( A ′ ∪ B ) ∼ = Pc( A ′ ) × Pc( B ) andby similar arguments we deduce that Pc( A ∪ B ′ ) ∼ = Pc( A ) × Pc( B ′ ). (cid:3) Orbits of essential roots: affine versus non-affine.
Using the GridLemma, we can now establish a basic description of the w -orbit of a w -essentialwall for some fixed w ∈ W . As before, we let W < W be a torsion-free finiteindex normal subgroup. Recall from Lemma 2.5 that for all n > w ) = Ess( w n ) and, moreover, the set Ess( w ) has finitely many orbits underthe action of h w i (and hence also under h w n i ). Proposition 2.10.
Let w ∈ W be of infinite order, let k > be such that w k ∈ W and let Ess( w ) = Ess( w k ) = M ∪ · · · ∪ M t be the partition of Ess( w ) into h w k i -orbits. For each i ∈ { , . . . , t } , let also P i = Pc( { r m | m ∈ M i } ) .Then for all i ∈ { , . . . , t } , the group P i is an irreducible direct component of Pc( w ) . In particular, for all j = i , we have either P i = P j or Pc( P i ∪ P j ) ∼ = P i × P j .More precisely, one of the following assertions holds.(i) P i = P j and each m ∈ M i meets finitely many walls in M j .(ii) P i = P j is irreducible affine.(iii) Pc( P i ∪ P j ) ∼ = P i × P j . Proof . Let i ∈ { , . . . , t } . Since M i is h w k i -invariant, it follows that P i is normal-ized by w k . As h r m | m ∈ M i i is an irreducible reflection group by Lemma 2.2(ii), P i is of irreducible non-spherical type by Lemma 2.2(iii). It then follows fromLemma 2.1 that N ( P i ) = P i × Z ( P i ) is itself a parabolic subgroup. In particularit contains Pc( w k ). Since on the other hand we have P i ≤ Pc( w k ) by Lemma 2.7,we infer that P i is a direct component of Pc( w k ). Since Pc( w k ) = Pc ∞ ( w ) isthe essential component of Pc( w ) by Lemma 2.7, we deduce that P i is a directcomponent of Pc( w ) as desired.Let now j = i . Since we already know that P i and P j are irreducible directcomponents of Pc( w ), it follows that either P i = P j or (iii) holds. So assume that P i = P j and that there exists a wall m ∈ M i meeting infinitely many walls in M j .We have to show that (ii) holds.Let λ be a w -axis. By Lemma 2.5, all walls in M i ∪ M j are transverse to λ .Moreover, by Lemma 2.6 the elements of M i (resp. M j ) are pairwise parallel.Therefore, we deduce that infinitely many walls in M i meet infinitely many wallsin M j . Since M i and M j are both h w k i -invariant, it follows that all walls in M i meet all walls in M j . Thus M i ∪ M j forms a grid and the desired conclusion followsfrom Lemma 2.8. (cid:3) We shall now deduce a rather subtle, but nevertheless important, differencebetween the affine and non-affine cases concerning the h w i -orbit of a w -essentialroot α . P-E. CAPRACE AND T. MARQUIS
Let us start by considering a specific example, namely the Coxeter group W = h r a , r b , r c i of type ˜ A , acting on the Euclidean plane. One verifies easily that W contains a nonzero translation t which preserves the r a -invariant wall m a . Let w = tr a . Then w is of infinite order so that Pc( w ) = W . Moreover the walls m b and m c , respectively fixed by r b and r c , are both w -essential by Lemma 2.5. Nowwe observe that, for each even integer n the walls m b and w n m b are parallel, whilefor each odd integer the walls m b and w n m b have a non-empty intersection.The following result (in the special case m = m ′ ) shows that the situation wehave just described cannot occur in the non-affine case. Proposition 2.11.
Let w ∈ W , m be a w -essential wall and P be the irreduciblecomponent of Pc( w ) that contains r m .If P is not of affine type, then for each w -essential wall m ′ such that r m ′ ∈ P ,there exists an l ∈ N such that for all l ∈ Z with | l | ≥ l , the wall m ′ lies between w − l m and w l m . Proof . First notice that if m is a w -essential wall, then the reflection r m belongsto Pc( w ) by Lemma 2.7, so that P is well defined. Moreover, we have r w l m = w l r m w − l ∈ P for all l ∈ Z .Let k > w k ∈ W and let Ess( w ) = Ess( w k ) = M ∪ · · · ∪ M t bethe partition of Ess( w ) into h w k i -orbits. Upon reordering the M i , we may assumethat m ′ ∈ M . Let also I ⊆ { , . . . , t } be the set of those i such that w l m ∈ M i for some l . In other words the h w i -orbit of m coincides with S i ∈ I M i .For all j , set P j = Pc( { r µ | µ ∈ M j } ). By Proposition 2.10, each P j is anirreducible direct component of Pc( w ). By hypothesis, this implies that P = P = P i for all i ∈ I .Suppose now that for infinitely many values of l , the wall w l m has a non-emptyintersection with m ′ . We have to deduce that P is of affine type.Recall from Lemma 2.6 that the elements of M j are pairwise parallel for all j .Therefore, our assumption implies that for some i ∈ I , the wall m ′ meets infinitelymany walls in M i . By Proposition 2.10, this implies that either P = P = P i isof affine type, or Pc( P ∪ P i ) ∼ = P × P i . The second case is impossible since P = P i . (cid:3) On parabolic closures of a pair of reflections.
The following conse-quence of Proposition 2.10 was stated as Theorem E in the introduction.
Corollary 2.12.
For each w ∈ W with infinite irreducible parabolic closure Pc( w ) ,there is a constant C such that the following holds. For all m, m ′ ∈ Ess( w ) with d ( m, m ′ ) > C , we have Pc( w ) = Pc( r m , r m ′ ) . We shall use the following.
Lemma 2.13.
Let α, β, γ ∈ Φ such that α ( β ( γ . Then r β ∈ Pc( { r α , r γ } ) . Proof . See [Cap06, Lemma 17]. (cid:3)
Proof of Corollary 2.12.
Retain the notation of Proposition 2.10. Since P =Pc( w ) is irreducible, we have P = P i for all i ∈ { , . . . , t } by Proposition 2.10.Recall that M i is the h w k i -orbit of some w -essential wall m . For all n ∈ Z , we set m n = w kn m . By Lemma 2.6 the elements of M i are pairwise parallel and hencefor all i < j < n , it follows that m j separates m i from m n . For all n ≥ Q n = Pc( { r m n , r m − n } ). By Lemma 2.13 we have Q n ≤ Q n +1 ≤ P for all n ≥
0. In
PEN SUBGROUPS OF KAC–MOODY GROUPS 9 particular S n ≥ Q n is a parabolic subgroup, which must thus coincide with P . Itfollows that Q n = P for some n . Since this argument holds for all i ∈ { , . . . , t } ,the desired result follows. (cid:3) Corollary 2.14.
Any irreducible non-spherical parabolic subgroup P is the para-bolic closure of a pair of reflections. Proof . Let w ∈ P such that P = Pc( w ). Such an w always exists by [CF10,Cor.4.3]. (Note that this can also be deduced from Corollary 2.17 below togetherwith [AB08, Prop.2.43].) The conclusion now follows from Corollary 2.12. (cid:3) The parabolic closure of a product of two elements in a Coxetergroup.
We are now able to present the main result of this section, which wasstated as Theorem G in the introduction.Before we state it, we prove one more technical lemma about CAT(0) spaces.Recall that W acts on the CAT(0) space X . For a hyperbolic w ∈ W , let | w | denote its translation length and set Min( w ) = { x ∈ X | d( x, wx ) = | w |} . Lemma 2.15.
Let w ∈ W be hyperbolic and suppose it decomposes as a product w = w w . . . w t of pairwise commuting hyperbolic elements of W . Let m be a w -essential wall. Then m is also w i -essential for some i ∈ { , . . . , t } . Proof . Write w := w . Then, since the w i are pairwise commuting for i =0 , . . . , t , each w i stabilizes Min( w j ) for all j . Thus M := T tj =1 Min( w j ) andMin( w ) are both non-empty by CAT(0)-convexity, and are stabilized by each w i , i = 0 , . . . , t . Therefore, if x ∈ M ∩ Min( g ), there is a piecewise geodesic path x, w x, w w x, . . . , w . . . w t x = wx inside M ∩ Min( g ), where each geodesic seg-ment is part of a w i -axis for some i ∈ { , . . . , t } . Since any wall intersecting thegeodesic segment [ x, wx ] must intersect one of those axis, the conclusion followsfrom Lemma 2.5. (cid:3) Theorem 2.16.
For all g, h ∈ W , there exists a constant K = K ( g, h ) ∈ N suchthat for all m, n ∈ Z with min {| m | , | n | , | m/n | + | n/m |} ≥ K , we have Pc( g ) ∪ Pc( h ) ⊆ Pc( g m h n ) . Proof . Fix g, h ∈ W . Let Ess( g ) = M ∪ · · · ∪ M k (resp. Ess( h ) = N ∪ · · · ∪ N l )be the partition of Ess( g ) into h g i -orbits (resp. Ess( h ) into h h i -orbits). For all i ∈ { , . . . , k } and j ∈ { , . . . , l } , set P i = Pc( { r m | m ∈ M i } ) and Q j =Pc( { r m | m ∈ N j } ).By Lemma 2.7, we have Pc( g ) = h{ r m | m ∈ M i , i = 1 , . . . , k }i , and Propo-sition 2.10 ensures that P i is an irreducible direct component of Pc( g ) for all i .Thus there is a subset I ⊆ { , . . . , k } such that Pc( g ) = Q i ∈ I P i . Similarly, thereis a subset J ⊆ { , . . . , l } such that Pc( h ) = Q j ∈ J Q j .For all i ∈ I and j ∈ J , we finally let g i and h j denote the respective projectionsof g and h onto P i and Q j , so that P i = Pc( g i ) and Q j = Pc( h j ).We define a collection E( g, h ) of subsets of W as follows: a set Z ⊆ W belongsto E( g, h ) if and only if there exists a constant K = K ( g, h, Z ) ∈ N such that forall m, n ∈ Z with min {| m | , | n | , | m/n | + | n/m |} ≥ K we have Z ⊆ Pc( g m h n ).Our goal is to prove that Pc( g ) and Pc( h ) both belong to E ( g, h ). To this end,it suffices to show that P i and Q j belong to E( g, h ) for all i ∈ I and j ∈ J . Thiswill be achieved in Claim 6 below. Claim 1. M s ⊆ Ess( g i ) for all s ∈ { , . . . , k } and i ∈ I such that P s = P i .Similarly, N s ⊆ Ess( h j ) for all s ∈ { , . . . , l } and j ∈ J such that Q s = Q j . Indeed, let m ∈ M s for some s ∈ { , . . . , k } . Then r m ∈ P s = P i . Moreover, as m is g -essential, it must be g i ′ -essential for some i ′ ∈ I by Lemma 2.15. But then r m ∈ Pc( g i ′ ) = P i ′ and so i ′ = i . The proof of the second statement is similar. Claim 2. If i ∈ I is such that [ P i , Q j ] = 1 for all j ∈ J , then P i belongs to E( g, h ) .Similarly, if j ∈ J is such that [ P i , Q j ] = 1 for all i ∈ I , then Q j belongs to E( g, h ) . Indeed, suppose [ P i , Q j ] = 1 for some i ∈ I and for all j ∈ J . Then P i commutes with Pc( h ). Thus h fixes every wall of M i . In particular, any wall µ ∈ M i is g m h n -essential for all m, n ∈ Z ∗ since g m h n = g mi w for some w ∈ W fixing µ and commuting with g i . Therefore P i ⊆ Pc( g m h n ) for all m, n ∈ Z ∗ andso P i belongs to E( g, h ). The second statement is proven in the same way. Claim 3.
Let i ∈ I and j ∈ J be such that P i = Q j . Then, for all m, n ∈ Z ,every g mi h nj -essential root is also g m h n -essential. Indeed, take α ∈ Φ and k > g mi h nj ) k α ( α . Notice that Pc( g mi h nj ) ⊆ P i = Q j , and hence r α ∈ P i = Q j by Lemma 2.7. Moreover, setting g ′ := Q t = i g mt and h ′ := Q t = j h nt , we have g ′ α = α = h ′ α since g ′ and h ′ centralize P i = Q j .Therefore ( g m h n ) k α = ( g mi h nj ) k ( g ′ h ′ ) k α = ( g mi h nj ) k α ( α so that α is also g m h n -essential. Claim 4.
Let i ∈ I and j ∈ J be such that P i = Q j . If P i is of affine type, then P i = Q j belongs to E( g, h ) . Since P i = Q j is of irreducible affine type, we have Pc( w ) = P i for all w ∈ P i of infinite order. Thus, in order to prove the claim, it suffices to show that thereexists some constant K such that g mi h nj is of infinite order for all m, n ∈ Z withmin {| m | , | n | , | m/n | + | n/m |} ≥ K . Indeed, we will then get that Pc( g mi h nj ) = P i isof essential type and so P i = Pc( g mi h nj ) ≤ Pc( g m h n ) by Claim 3 and Lemma 2.7(ii).Recalling that P i is of affine type, we can argue in the geometric realization ofa Coxeter complex of affine type, which is a Euclidean space. We deduce thatif g i and h j have non-parallel translation axes, then g mi h nj is of infinite order forall nonzero m, n . On the other hand, if g i and h j have some parallel translationaxes, we consider a Euclidean hyperplane H orthogonal to these and let ℓ i and ℓ j denote the respective translation lengths of g i and h j . Then, upon replacing g i by its inverse (which does not affect the conclusion since E ( g, h ) = E ( g − , h )), wehave d ( g mi h nj H, H ) = | mℓ i − nℓ j | . Since g mi h nj is of infinite order as soon as thisdistance is nonzero, the claim now follows by setting K = ℓ i /ℓ j + ℓ j /ℓ i + 1. Claim 5.
Let i ∈ { , . . . , k } and j ∈ { , . . . , l } be such that M i ∩ N j is infinite.Then P i = Q j and these belong to E( g, h ) . Indeed, remember that the walls in M i are pairwise parallel by Lemma 2.6.Since M i ∩ N j ⊆ Ess( g i ′ ) ∩ Ess( h j ′ ) for some i ′ ∈ I such that P i = P i ′ and some j ′ ∈ J such that Q j = Q j ′ by Claim 1, Corollary 2.12 then yields P i = Q j .Let now C denote the minimal distance between two parallel walls in X and set K := | g | + | h | C + 1. Let m, n ∈ Z be such that min {| m | , | n | , | m/n | + | n/m |} ≥ K . We PEN SUBGROUPS OF KAC–MOODY GROUPS 11 now show that P i ≤ Pc( g m h n ). By Lemma 2.7 and Corollary 2.12, it is sufficientto check that infinitely many walls in M i ∩ N j are g m h n -essential.Note first that for any wall µ ∈ M i ∩ N j , we have g ǫm µ ∈ M i and h ǫn µ ∈ N j for ǫ ∈ { + , −} . Thus, since M i ∩ N j is infinite, there exist infinitely many such µ ∈ M i ∩ N j with the property that g ǫm µ lies between µ and some µ ǫ ∈ M i ∩ N j and h ǫn µ lies between µ and some µ ′ ǫ ∈ M i ∩ N j for ǫ ∈ { + , −} . We now showthat any such µ is g m h n -essential, as desired. Consider thus such a µ .Let D be a g -axis and D ′ be an h -axis. Since M i ∩ N j ⊆ Ess( g ) ∩ Ess( h ),Lemma 2.5 implies that each of the walls µ , µ ǫ and µ ′ ǫ for ǫ ∈ { + , −} is transverseto both D and D ′ . In particular, the choice of µ implies that g ǫm µ and h ǫn µ for ǫ ∈ { + , −} are also transverse to both D and D ′ .Let α ∈ Φ be such that ∂α = µ and g m α ( α . If h n α ( α then clearly g m h n α ( α , as desired. Suppose now that h n α ) α .Note that the walls in h g i µ ∪ h h i µ are pairwise parallel since this is the case forthe walls in W · µ by Lemma 2.6 and since g, h ∈ W .Assume now that | n | > | m | , the other case being similar. In particular, | n/m | > | g | /C . Then d( µ, g − m µ ) ≤ | m | · | g | < | n | · C ≤ d( µ, h n µ ) and so the wall g − m µ liesbetween µ and h n µ . Thus α ( g − m α ( h n α and so g m h n α ) α , as desired. Claim 6.
For all i ∈ I and j ∈ J , the sets P i and Q j both belong to E( g, h ) . We only deal with P i ; the argument for Q j is similar.Let D denote a g -axis, and D ′ an h -axis in X . By Claim 5 we may assume that M i ∩ Ess( h ) is finite. Moreover, by Claim 3 we may assume there exists a j ∈ J such that [ P i , Q j ] = 1.If N j ∩ Ess( g ) is infinite, then N j ∩ M i ′ is infinite for some i ′ ∈ { , . . . , k } andthus Claim 5 yields that Q j = P i ′ ∈ E( g, h ). In particular, [ Q j , P s ] = 1 as soonas P s = P i ′ . This implies P i = P i ′ ∈ E( g, h ), as desired. We now assume that N j ∩ Ess( g ) is finite.Thus by Lemma 2.5, only finitely many walls in M i intersect D ′ and only finitelywalls in N j intersect D .Take m ∈ M i and m ∈ N j . By Claim 1 and Corollary 2.12, there exists some k ∈ N such that if one sets M := { g sk m | s ∈ Z } ⊆ M i and N := { h tk m | t ∈ Z } ⊆ N j , then any two reflections associated to distinct walls of M (respectively, N ) generate P i (respectively, Q j ) as parabolic subgroups. Also, we may assumethat no wall in M intersects D ′ and that no wall in N intersects D .If every wall of M intersects every wall of N , then since [ P i , Q j ] = 1, Lemma 2.8yields that P i = Q j is of affine type and Claim 4 allows us to conclude. Up tomaking a different choice for m and m inside M and N respectively, we maythus assume that m is parallel to m . For the same reason, we may also choose m ′ ∈ M and m ′ ∈ N such that D ′ lies between m and m ′ , D lies between m and m ′ , and such that m ∩ m ′ = m ∩ m ′ = m ′ ∩ m ′ = ∅ .Let now s , t ∈ Z be such that g s k m = m ′ and h t k m = m ′ . Up tointerchanging m and m ′ (respectively, m and m ′ ), we may assume that s > t > α, β ∈ Φ be such that ∂α = m , ∂β = m and such that D ′ is contained in α ∩ − g s k α and D is contained in β ∩ − h t k β . For each s, t ∈ Z , set α s := g sk α and β t = h tk β (see Figure 1). Since for two roots γ, δ ∈ Φ with ∂γ parallel to ∂δ ,one of the possibilities γ ⊆ δ or γ ⊆ − δ or − γ ⊆ δ or − γ ⊆ − δ must hold, this g hα s α s +1 β t β t +1 α = α α − β = β β − Figure 1.
Claim 6.implies that α s ⊆ − β t , − α ⊆ β and β t ⊆ α. Set K := ( s + t + 1) k and let m, n ∈ Z be such that | m | , | n | > K . We nowprove that P i ≤ Pc( g m h n ). By Lemma 2.7, it is sufficient to show that either α − and α or α s and α s +1 are g m h n -essential. We distinguish several cases dependingon the respective signs of m, n . • If m, n >
0, then g m h n α s +1 ⊆ g m h n α s ⊆ g m h n β ( g m β t ⊆ g m α ( α s +1 ⊆ α s so that α s and α s +1 are g m h n -essential. • If m, n <
0, then g m h n α − ⊇ g m h n α ⊇ g m h n β t ) g m β ⊇ g m α s ) α − ⊇ α so that α − and α are g m h n -essential. • If m > n <
0, then g m h n α s +1 ⊆ g m h n α s ⊆ g m h n ( − β t ) ( g m ( − β ) ⊆ g m α ( α s +1 ⊆ α s so that α s and α s +1 are g m h n -essential. • If m < n >
0, then g m h n α − ⊇ g m h n α ⊇ g m h n ( − β ) ) g m ( − β t ) ⊇ g m α s ) α − ⊇ α so that α − and α are g m h n -essential.This concludes the proof of the theorem. (cid:3) The following corollary will be of fundamental importance in the rest of thepaper. It was stated as Corollary H in the introduction.
Corollary 2.17.
Let H be a subgroup of W . Then there exists h ∈ H ∩ W suchthat [Pc( H ) : Pc( h )] < ∞ . Proof . Take h ∈ H ∩ W such that Pc( h ) is maximal. Then Pc( h ) = Pc( H ∩ W ), for otherwise there would exist g ∈ H ∩ W such that Pc( g ) Pc( h ),and hence Theorem 2.16 would yield integers m, n such that Pc( h ) ( Pc( g m h n ),contradicting the choice of h . The result now follows from Lemma 2.4 since[ H : H ∩ W ] < ∞ . (cid:3) PEN SUBGROUPS OF KAC–MOODY GROUPS 13
Remark 2.18.
Note that the conclusion of Corollary 2.17 cannot be improved:indeed, one cannot expect that there is some h ∈ H such that Pc( H ) = Pc( h )in general. Consider for example the Coxeter group W = h s i × h t i × h u i , whichis a direct product of three copies of Z / Z . Then the parabolic closure of thesubgroup H = h st, tu i of W is the whole of W , but there is no h ∈ H such thatPc( h ) = W .2.9. On walls at bounded distance from a residue.
We finish this sectionwith a couple of observations on Coxeter groups which we shall need in our studyof open subgroups of Kac–Moody groups.Given a subset J ⊆ S , we set Φ J = { α ∈ Φ | ∃ v ∈ W J , s ∈ J : α = vα s } , where α s denotes the positive root associated with the reflection s . Lemma 2.19.
Let L ⊆ S be essential. Then for each root α ∈ Φ L , there exists w ∈ W L such that w.α ( α . In particular α is w -essential. Proof . Let α ∈ Φ L . By [H´ee93, Prop. 8.1, p. 309], there exists a root β ∈ Φ L suchthat α ∩ β = ∅ . We can then take w = r α r β or its inverse. (cid:3) Lemma 2.20.
Let L ⊆ S be essential, and let R be the standard L -residue of theCoxeter complex Σ of W .Then for each wall m of Σ , the following assertions are equivalent: (i) m is perpendicular to every wall of R , (ii) [ r m , W L ] = 1 , (iii) There exists n > such that R is contained in an n -neighbourhood of m . Proof . We first show that (iii) ⇒ (ii). By Lemma 2.19, if m ′ is a wall of R (thatis, a wall intersecting R ), then there exists w ∈ W L such that one of the twohalf-spaces associated to m ′ is w -essential. It follows that m and m ′ cannot beparallel since R is at a bounded distance from m . Hence m is transversal to everywall of R , and does not intersect R . Back to an arbitrary wall m ′ of R , considera wall m ′′ of R that is parallel to m ′ and such that the reflection group generatedby the two reflections r m ′ and r m ′′ is infinite dihedral. Such a wall m ′′ exists byLemma 2.19. Then r m centralizes these reflections by Lemma 2.3 and [CR09,Lem.12]. As m ′ was arbitrary, this means that r m centralizes W L .The equivalence of (i) and (ii) is trivial.Finally, to show (i) ⇒ (iii), notice that if C is a chamber of R and t a reflectionassociated to a wall of R , then the distance from C to m equals the distance from t · C to m . Indeed, if α is the root associated to m not containing R and D isthe projection of C onto α , then t · D is the projection of t · C onto α . As W L istransitive on R , (iii) follows. (cid:3) Open and parabolic subgroups of Kac–Moody groups
Basics on Kac–Moody groups and their completions can be found in [R´em02],[CR09] and references therein. We focus here on the case of a finite ground field.Let G = G ( F q ) be a (minimal) Kac–Moody group over a finite field F q of order q . The group G is endowed with a root group datum { U α | α ∈ Φ = Φ(Σ(
W, S )) } for some Coxeter system ( W, S ), which yields a twin BN-pair ( B + , B − , N ) withassociated twin building (∆ + , ∆ − ). Let C be the fundamental chamber of ∆ + , namely the chamber such that B + = Stab G ( C ), and let A ⊂ ∆ + be the funda-mental apartment, so that N = Stab G ( A ) and H := B + ∩ N = Fix G ( A ). Weidentify Φ with the set of half-spaces of A .We next let G be the completion of G with respect to the positive buildingtopology. Thus the finitely generated group G embeds densely in the topologicalgroup G , which is locally compact, totally disconnected and acts properly andcontinuously on ∆ := ∆ + by automorphims. A completed Kac–Moody groupover a finite field shall be called a locally compact Kac–Moody group . Let B = B + be the closure of B + in G , let N = Stab G ( A ) and H = B ∩ N = Fix G ( A ).[We warn the reader that N and H are discrete, whence closed in G while N and H are non-discrete closed subgroups.] The pair ( B, N ) is a BN-pair of type (
W, S )for G ; in particular we have N/H ∼ = W . Moreover, the group B is a compact opensubgroup, and every standard parabolic subgroup P J = BW J B for some J ⊆ S is thus open in G . Important to our later purposes is the fact that the group G acts transitively on the complete apartment system of ∆. In particular B actstransitively on the apartments containing C .For a root α ∈ Φ, we denote as before the unique reflection of W fixing the wall ∂α pointwise by r α . In addition, we choose some element n α ∈ N ∩ h U α ∪ U − α i which maps onto r α under the quotient map N → N/H ∼ = W .Before we state a more precise version of Theorem A, we will need some addi-tional results on the BN-pair structure of G . This is the object of the followingparagraph.3.1. On Levi decompositions in complete Kac–Moody groups.
Given J ⊆ S , we denote by P J = B + W J B + (resp. P J = BW J B ) the standard parabolicsubgroup of G (resp. G ) of type J and by R J ( C ) the J -residue of ∆ containingthe chamber C . Thus P J = Stab G ( R J ( C )), P J = Stab G ( R J ( C )) and P J is densein P J .We further set Φ J = { α ∈ Φ | ∃ v ∈ W J , s ∈ J : α = vα s } and L + J = h U α | α ∈ Φ J i . Finally, we set L J = H · L + J and denote by U J the normal closure of h U α | α ∈ Φ , α ⊃ R J ( C ) ∩ A i in B + . Following [R´em02, 6.2.2], there is a semidirectdecomposition P J = L J ⋉ U J . The group U J is called the unipotent radical of the parabolic subgroup P J , and L J is called the Levi factor .We next define L + J = L + J , L J = L J and U J = U J . Thus U J and L J are closed subgroups of P J , respectively called the unipotentradical and the Levi factor . Lemma 3.1.
We have the following:(i) U J is a compact normal subgroup of P J , and we have P J = L J · U J .(ii) L + J is normal in L J and we have L J = H · L + J . Proof . Since U J is normal in P J , which is dense in P J , it is clear that U J is normalin P J . Moreover U J is compact (since it is contained in B ) and the product L J · U J is thus closed in P J . Assertion (i) follows since L J · U J contains P J . PEN SUBGROUPS OF KAC–MOODY GROUPS 15
For assertion (ii), we remark that H normalizes L + J and hence also L + J . More-over, since H is finite, hence compact, the product H · L + J is closed. Since H · L + J is dense in L J , the conclusion follows. (cid:3) Remark that the decomposition P J = L J · U J is even semidirect when J isspherical, see [RR06, section 1.C.]. It is probably also the case in general, but thiswill not be needed here. Lemma 3.2.
Let J ⊆ S . Then every open subgroup O of P J that contains theproduct L + J · U J ∪ J ⊥ has finite index in P J . Proof . Set K := J ⊥ and U := U J ∪ J ⊥ . Note that U ⊳ P J ∪ K = L J ∪ K · U . Moreover, L + J is normal in L J ∪ K . Indeed, as [ U α , U β ] = 1 for all α ∈ Φ J and β ∈ Φ K , thesubgroups L + J and L + K centralize each other. Since in addition H normalizes eachroot group, we get a decomposition L J ∪ K = H · L + J · L + K . In particular, L J ∪ K normalizes L + J , whence also L + J . As the normalizer of a closed subgroup is closed,this implies that L J ∪ K normalizes L + J , as desired.Let π : P J ∪ K → P J ∪ K /U denote the natural projection. Then π ( L + J ) is normalin P J ∪ K /U , since it is the image of L + J under the composition map L J ∪ K → L J ∪ K L J ∪ K ∩ U ∼ = → P J ∪ K U : l l ( L J ∪ K ∩ U ) lU. Let π : P J ∪ K → π ( P J ∪ K ) /π ( L + J ) denote the composition of π with the canon-ical projection onto π ( P J ∪ K ) /π ( L + J ). Note that π is an open continuous grouphomomorphism. Then π ( P J ) = π ( L + J · U J · H ) /π ( L + J ) is compact. Indeed, it ishomeomorphic to the quotient of the compact group π ( U J · H ) by the normalsubgroup π ( L + J ∩ U J · H ) under the map π ( L + J · U J · H ) π ( L + J ) ∼ = → π ( U J · H ) π ( L + J ∩ U J · H ) : π ( l · u ) π ( L + J ) π ( u ) π ( L + J ∩ U J · H ) . In particular, since π ( O ) is open in π ( P J ), it has finite index in π ( P J ). But thensince O = π − ( π ( O )) by hypothesis, O has finite index in π − ( π ( P J )) = P J , asdesired. (cid:3) A refined version of Theorem A.
We will prove the following statement,having Theorem A as an immediate corollary.
Theorem 3.3.
Let O be an open subgroup of G . Let J ⊆ S be the type of aresidue which is stabilized by some finite index subgroup of O and minimal withrespect to this property.Then there exist a spherical subset J ′ ⊆ J ⊥ and an element g ∈ G such that L + J · U J ∪ J ⊥ < gOg − < P J ∪ J ′ . In particular, gOg − has finite index in P J ∪ J ′ .Moreover, any subgroup of G containing gOg − as a finite index subgroup iscontained in P J ∪ J ′′ for some spherical subset J ′′ ⊆ J ⊥ . In particular, only finitelymany distinct parabolic subgroups contain O as a finite index subgroup. Proof of Theorem 3.3: outline and first observations.
This sectionand the next ones are devoted to the proof of Theorem 3.3 itself.Let thus O be an open subgroup of G . We define the subset J of S as in thestatement of the theorem, namely, J is minimal amongst the subsets L of S forwhich there exists a g ∈ G such that O ∩ g − P L g has finite index in O . For sucha g ∈ G , we set O = gOg − ∩ P J . Thus O stabilizes R J ( C ) and is an opensubgroup of G contained in gOg − with finite index.We first observe that the desired statement is essentially empty when O iscompact. Indeed, in that case the Bruhat–Tits fixed point theorem ensures that O stabilizes a spherical residue of G , and hence Theorem 3.3 stands proven with J = ∅ . It thus remains to prove the theorem when O , and hence also O , isnon-compact, which we assume henceforth.Recall from the previous section that we call a subset J ⊆ S essential if allits irreducible components are non-spherical. We begin with the following simpleobservation. Lemma 3.4. J is essential. Proof . Let J ⊆ J denote the union of the non-spherical irreducible componentsof J . As P J has finite index in P J , the subgroup O ∩ P J is open of finite indexin O and stabilizes R J ( C ). The definition of J then yields J = J . (cid:3) Let us now describe the outline of the proof. Our first task will be to showthat O contains L + J . We will see that this is equivalent to prove that O actstransitively on the standard J -residue R J ( C ), or else that the stabilizer in O of any apartment A containing C is transitive on R J ( C ) ∩ A . Since each groupStab O ( A ) / Fix O ( A ) can be identified with a subgroup of the Coxeter group W acting on A , we will be in a position to apply the results on Coxeter groups fromthe previous section. This will allow us to show that each Stab O ( A ) / Fix O ( A )contains a finite index parabolic subgroup of type I A ⊆ J , and hence acts transi-tively on the corresponding residue.We thus begin by defining some “maximal” subset I of J such that Stab O ( A )acts transitively on R I ( C ) ∩ A for a suitably chosen apartment A containing C . We then establish that I contains all the types I A when A varies over allapartments containing C . This eventually allows us to prove that in fact I = J ,so that Stab O ( A ) is transitive on R J ( C ) ∩ A , or else that O contains L + J , asdesired.We next show that O contains the unipotent radical U J ∪ J ⊥ . Finally, we makeuse of the transitivity of O on R J ( C ) to prove that O is contained in the desiredparabolic subgroup.3.4. Proof of Theorem 3.3: O contains L + J . We first need to introduce someadditional notation which we will retain until the end of the proof.Let A ≥ C denote the set of apartments of ∆ containing C . For A ∈ A ≥ C , set N A := Stab O ( A ) and N A = N A / Fix O ( A ), which one identifies with a subgroupof W . Finally, for h ∈ N A , denote by h its image in N A ≤ W . Here is the maintool developed in the previous section. PEN SUBGROUPS OF KAC–MOODY GROUPS 17
Lemma 3.5.
For all A ∈ A ≥ C , there exists h ∈ N A such that Pc( h ) = h r α | α h -essential root of Φ i and is of finite index in Pc( N A ) . Proof . This is an immediate consequence of Corollary 2.17 and Lemma 2.7. (cid:3)
Lemma 3.6.
Let ( g n ) n ∈ N be an infinite sequence of elements of O . Then thereexist an apartment A ∈ A ≥ C , a subsequence ( g ψ ( n ) ) n ∈ N and elements z n ∈ O , n ∈ N , such that for all n ∈ N we have (1) h n := z − z n ∈ N A , (2) d( C , z n R ) = d( C , g ψ ( n ) R ) for every residue R containing C and (3) | d( C , h n C ) − d( C , g ψ ( n ) C ) | < d( C , z C ) . Proof . As O is open, it contains a finite index subgroup K := Fix G ( B ( C , r ))of B for some r ∈ N . Since B is transitive on the set A ≥ C , we deduce that K has only finitely many orbits in A ≥ C , say A , . . . , A k . So, up to choosinga subsequence, we may assume that all chambers g n C belong to the same K -orbit A i of apartments. Hence there exist elements x n ∈ K ⊂ O and anapartment A ′ ∈ A i containing C such that g ′ n := x n g n ∈ O , g ′ n C ∈ A ′ andd( C , g ′ n C ) = d( C , g n C ). For each n , we now choose an element of G stabilizing A ′ and mapping C to g ′ n C . Thus such an element is in the same right cosetmodulo B as g ′ n . In particular, up to choosing a subsequence, we may assume ithas the form g ′ n y n b ∈ Stab G ( A ′ ) for some y n ∈ K and some b ∈ B independantof n . Denote by { ψ ( n ) | n ∈ N } the resulting indexing set for the subsequence.Then setting A := bA ′ ∈ A ≥ C , the sequence z n := g ′ ψ ( n ) y ψ ( n ) ∈ O is such that h n := z − z n ∈ b Stab G ( A ′ ) b − ∩ O = Stab O ( A ) = N A and | d( C , h n C ) − d( C , g ψ ( n ) C ) | = | d( z C , z n C ) − d( C , z n C ) | < d( C , z C ) . (cid:3) Lemma 3.7.
There exists an apartment A ∈ A ≥ C such that the orbit N A · C isunbounded. In particular, the parabolic closure in W of N A is non-spherical. Proof . Since O is non-compact, the orbit O · C is unbounded in ∆. For n ∈ N ,choose g n ∈ O such that d( C , g n C ) ≥ n . Then by Lemma 3.6, there exist anapartment A ∈ A ≥ C and elements h n ∈ N A for n in some unbounded subsetof N such that d( C , h n C ) is arbitrarily large when n varies. This proves thelemma. (cid:3) Let A ∈ A ≥ C be an apartment such that the type of the product of the non-spherical irreducible components of Pc( N A ) is nonempty and maximal for thisproperty. Such an apartment exists by Lemma 3.7. Now choose h A ∈ N A as inLemma 3.5, so that in particular [Pc( N A ) : Pc( h A )] < ∞ . Up to conjugating O by an element of P J , we may then assume without loss of generality that Pc( h A )is standard, non-spherical, and has essential type I . Moreover, it is maximal in thefollowing sense: if A ∈ A ≥ C is such that Pc( N A ) contains a parabolic subgroupof essential type I A with I A ⊇ I , then I = I A .Now that I is defined, we need some tool to show that O contains sufficientlymany root groups U α . This will ensure that O is “transitive enough” in two ways:first on residues in the building by showing it contains subgroups of the form L + T ,and second on residues in apartments by establishing the presence in O of enough n α ∈ h U α ∪ U − α i , since these lift reflections r α in stabilizers of apartments. Thistool is provided by the so-called (FPRS) property from [CR09, 2.1], which we nowstate. Note for this that as O is open, it contains the fixator in G of a ball of ∆:we fix r ∈ N such that O ⊃ K r := Fix G ( B ( C , r )). Lemma 3.8.
There exists a constant N = N ( W, S, r ) ∈ N such that for every root α ∈ Φ with d( C , α ) > N , the root group U − α is contained in Fix G ( B ( C , r )) = K r . Proof . See [CR09, Prop. 4]. (cid:3)
We also record a version of this result in a slightly more general setting.
Lemma 3.9.
Let g ∈ G and let A ∈ A ≥ C containing the chamber D := gC .Also, let b ∈ B such that A = bA , and let α = bα be a root of A , with α ∈ Φ .Then there exists N = N ( W, S, r ) ∈ N such that if d( D, − α ) > N then bU α b − ⊆ gK r g − . Proof . Take for N = N ( W, S, r ) the constant of Lemma 3.8 and suppose thatd( D, − α ) > N . Let h ∈ Stab G ( A ) be such that hC = b − D . Then N < d( D, − α ) = d( bhC , − bα ) = d( hC , − α ) = d( C , − h − α ) , and so Lemma 3.8 implies h − U α h = U h − α ⊆ K r . Let b ∈ B such that bh = gb .Then bU α b − ⊆ bhK r h − b − = gb K r b − g − = gK r g − . (cid:3) This will prove especially useful in the following form, when we will use thedescription of the parabolic closure of some w ∈ W in terms of w -essential rootsas in Lemma 3.5. Lemma 3.10.
Let A ∈ A ≥ C and b ∈ B such that A = bA . Also, let α = bα ( α ∈ Φ ) be a w -essential root of A for some w ∈ Stab G ( A ) / Fix G ( A ) , and let g ∈ Stab G ( A ) be a representative of w . Then there exists n ∈ Z such that for ǫ ∈ { + , −} we have U ǫα ⊆ b − g ǫn K r g − ǫn b . Proof . Choose n ∈ Z such that d( g ǫn C , − ǫα ) > N for ǫ ∈ { + , −} , where N = N ( W, S, r ) is the constant appearing in the statement of Lemma 3.8. Thus, for ǫ ∈ { + , −} we have d( b − g ǫn C , − ǫα ) > N, and so d( C , − ǫ ( b − g − ǫn b ) α ) > N .Lemma 3.8 then yields( b − g − ǫn b ) U ǫα ( b − g − ǫn b ) − = U ǫ ( b − g − ǫn b ) α ⊆ K r , and so U ǫα ⊆ ( b − g ǫn b ) K r ( b − g − ǫn b ) = ( b − g ǫn ) K r ( g − ǫn b ) . (cid:3) We are now ready to prove how the different transitivity properties of O arerelated. Lemma 3.11.
Let T ⊆ S be essential, and let A ∈ A ≥ C . Then the following areequivalent: (1) O contains L + T ; (2) O is transitive on R T ( C ) ; (3) N A is transitive on R T ( C ) ∩ A ; (4) N A contains the standard parabolic subgroup W T of W . PEN SUBGROUPS OF KAC–MOODY GROUPS 19
Proof . The equivalence (3) ⇔ (4), as well as the implications (1) ⇒ (2) , (3) aretrivial.To see that (4) ⇒ (2), note that if b ∈ B maps A onto A , then for each α ∈ Φ T , we have bU ± α b − ⊆ O , and so O ⊇ b L + T b − is transitive on R T ( C ).Indeed, let α ∈ Φ T and consider the corresponding root α := bα ∈ Φ T ( A ) of A . By Lemma 2.19, there exists w ∈ W T ⊆ N A such that α is w -essential. Thenif g ∈ O is a representative for w , Lemma 3.10 yields an n ∈ Z such that for ǫ ∈ { + , −} we have U ǫα ⊆ b − g ǫn K r g − ǫn b ⊆ b − O b .Finally, we show (2) ⇒ (1). Again, it is sufficient to check that if α ∈ Φ T ,then O contains U ǫα for ǫ ∈ { + , −} . By Lemma 2.19, there exists g ∈ Stab G ( A )stabilizing R T ( C ) ∩ A such that α is g -essential, where g denotes the image of g in the quotient group Stab G ( A ) / Fix G ( A ). Then, by Lemma 3.10, one can findan n ∈ Z such that U ǫα ⊆ g ǫn K r g − ǫn for ǫ ∈ { + , −} . Now, since O is transitiveon R T ( C ), there exist h ǫ ∈ O such that h ǫ C = g ǫn C , and so we find b ǫ ∈ B such that g ǫn = h ǫ b ǫ . Therefore U ǫα ⊆ h ǫ b ǫ K r b − ǫ h − ǫ = h ǫ K r h − ǫ ⊆ O . (cid:3) Now, to ensure that O indeed satisfies one of those properties for some “max-imal T ”, we use Lemma 3.5 to show that stabilizers in O of apartments containfinite index parabolic subgroups. Lemma 3.12.
Let A ∈ A ≥ C . Then there exists I A ⊆ S such that N A contains aparabolic subgroup P I A of W of type I A as a finite index subgroup. Proof . Choose h ∈ N A as in Lemma 3.5, so that in particular Pc( h ) is generatedby the reflections r α with α an h -essential root of A . Let α = bα be such a root( α ∈ Φ), where b ∈ B maps A onto A . By Lemma 3.10, we then find K ∈ Z such that for ǫ ∈ { + , −} , U ǫα ⊆ ( b − h ǫK ) K r ( h − ǫK b ) ⊆ b − O b. In particular, n α ∈ h U α ∪ U − α i ⊆ b − O b . As r α is the image in W of n α and since r α = br α b − , we finally obtain Pc( h ) ⊆ N A . Then P I A := Pc( h ) is thedesired parabolic subgroup, of type I A . (cid:3) For each A ∈ A ≥ C , we fix such an I A ⊆ S which, without loss of generality,we assume essential. We also consider the corresponding parabolic P I A containedin N A . Note then that P I A has finite index in Pc( N A ) by Lemma 2.4, and so I = I A . Lemma 3.13. O contains L + I . Proof . As noted above, we have I = I A and P I = W I . Since O is closed in G ,Lemma 3.11 allows us to conclude. (cid:3) We now have to show that I is “big enough”, that is, I = J . For this, we firstneed to know that I is “uniformly” maximal amongst all apartments containing C . Lemma 3.14.
Let A ∈ A ≥ C . Then I A ⊆ I . Proof . Set R := R I ( C ) ∩ A and let R be an I A -residue in A on which N A actstransitively and that is at minimal distance from R amongst such residues. Notethat N A is transitive on R as well by Lemma 3.11.If R ∩ R is nonempty, then N A is also transitive on the standard I ∪ I A -residueof A and so N A contains W I ∪ I A . By maximality of I and since I ∪ I A is againessential, this implies I A ⊆ I , as desired.We henceforth assume that R ∩ R = ∅ . Let b ∈ B such that bA = A .Consider a root α = bα of A , α ∈ Φ, whose wall ∂α separates R from R .If both R and R are at unbounded distance from ∂α , then the transitivity of N A on R and R together with Lemma 3.9 yield bU ± α b − ⊆ K r ⊆ O . Since r α ∈ h U α ∪ U − α i , we thus have r α := br α b − ∈ O and so r α ∈ N A . But then N A = r α N A r − α is also transitive on the I A -residue r α R which is closer to R , acontradiction.If R is at bounded distance from ∂α then by Lemma 2.20, r α centralizes thestabilizer P in W of R , that is, P = r α P r − α . Note that N A contains P since itis transitive on R . Thus N A is transitive on the I A -residue r α R , which is closerto R , again a contradiction.Thus we are left with the case where R is contained in a tubular neighbourhoodof every wall ∂α separating R from R . But in that case, Lemma 2.20 again yieldsthat W I is centralized by every reflection r α associated to such walls. Choosechambers C i in R i , i = 1 ,
2, such that d( C , C ) = d( R , R ), and let ∂α , . . . , ∂α k be the walls separating C from C , crossed in that order by a minimal gallery from C to C . Then each α i , 1 ≤ i ≤ k , separates R from R and so w := r α k . . . r α centralizes W I and maps C to C . So W I = wW I w − ⊆ N A is transitive on wR and R , and hence also on R I ∪ I A ( C ) ∩ A . Therefore N A contains a parabolicsubgroup of essential type I ∪ I A , so that I ⊇ I A by maximality of I , as desired. (cid:3) Lemma 3.15.
Let A ∈ A ≥ C . Then N A contains W I as a subgroup of finiteindex. Proof . We know by Lemmas 3.11 and 3.13 that N A contains W I . Also, byLemma 3.12, N A contains a finite index parabolic subgroup P I A = wW I A w − of type I A , for some w ∈ W . Since I A ⊆ I by Lemma 3.14, we get W I A ⊆ N A and so the parabolic subgroup P := W I A ∩ wW I A w − has finite index in W I A . As I A is essential, [AB08, Prop.2.43] then yields P = W I A and so W I A ⊆ wW I A w − .Finally, since the chain W I A ⊆ wW I A w − ⊆ w W I A w − ⊆ . . . stabilizes, we findthat W I A = P I A has finite index in N A . The result follows. (cid:3) We are now ready to make the announced connection between I and J . Lemma 3.16. I = J . Proof . Let R denote the set of I -residues of ∆ containing a chamber of O · C ,and set R := R I ( C ). We first show that the distance from C to the residues of R is bounded, and hence that R is finite.Indeed, suppose for a contradiction that there exists a sequence of elements g n ∈ O such that d( C , g n R ) ≥ n for all n ∈ N . Then, up to choosing asubsequence and relabeling, Lemma 3.6 yields an apartment A ∈ A ≥ C and asequence ( z n ) n ≥ n of elements of O such that h n := z − n z n ∈ N A and d( C , z n R ) =d( C , g n R ). Moreover by Lemma 3.15, we have a finite coset decomposition of theform N A = ` tj =1 v j W I . Denote by π : N A → N A the natural projection. Again PEN SUBGROUPS OF KAC–MOODY GROUPS 21 up to choosing a subsequence and relabeling, we may assume that π ( h n ) = v j u n for all n ≥ n (for some fixed n ∈ N ), where each u n ∈ W I and where j isindependant of n . Then the elements w n := π ( h − n h n ) = π ( z − n z n ) belong to W I .Thus the chambers z n C and z n C belong to the same I -residue since z n maps an I -gallery between C and w n C to an I -gallery between z n C and z n C . Therefored( C , g n R ) = d( C , z n R ) ≤ d( C , z n C )and so d( C , g n R ) is bounded, a contradiction.So R is finite and is stabilized by O . Hence the kernel O ′ of the inducedaction of O on R is a finite index subgroup of O stabilizing an I -residue. Up toconjugating by an element of O , we thus have O ′ < P I and [ O : O ′ ] < ∞ . Then O ′′ := O ∩ P I is open and contains O ′ , and has therefore finite index in O . Thedefinition of J finally implies that I = J . (cid:3) In particular, Lemmas 3.13 and 3.16 yield the following.
Corollary 3.17. O contains L + J . Proof of Theorem 3.3: O contains the unipotent radical U J ∪ J ⊥ . Toshow that O contains the desired unipotent radical, we again make use of the(FPRS) property. Lemma 3.18. O contains the unipotent radical U J ∪ J ⊥ . Proof . By definition of U J ∪ J ⊥ , we just have to check that for every b ∈ B andevery α ∈ Φ containing R J ∪ J ⊥ ( C ) ∩ A , we have bU α b − ∈ O . Fix such b and α . In particular, α contains R := R J ( C ) ∩ A . We claim that R is at unboundeddistance from the wall ∂α associated to α . Indeed, if it were not, then as J isessential by Lemma 3.4, the reflection r α would centralize W J by Lemma 2.20,and hence would belong to W J ⊥ by Lemma 2.1, contradicting α ⊃ R J ⊥ ( C ) ∩ A .Set now A = bA . Then α ′ = bα is a root of A containing R ′ := R J ( C ) ∩ A .Moreover, R ′ is at unbounded distance from − α ′ . Since O is transitive on R J ( C )by Corollary 3.17, there exists g ∈ O such that D := gC ∈ R J ( C ) ∩ A andd( D, − α ′ ) > N , where N is provided by Lemma 3.9. This lemma then impliesthat bU α b − ⊆ gK r g − ⊆ O , as desired. (cid:3) Proof of Theorem 3.3: endgame.
We can now prove that gOg − is con-tained in a parabolic subgroup that has P J as a finite index subgroup. Lemma 3.19.
Every subgroup H of G containing O as a subgroup of finite indexis contained in some standard parabolic P J ∪ J ′ of type J ∪ J ′ , with J ′ sphericaland J ′ ⊆ J ⊥ . Proof . Recall that O stabilizes the J -residue R := R J ( C ) and acts transitivelyon its chambers by Corollary 3.17. Let R be the (finite) set of J -residues of ∆containing a chamber in the orbit H · C .We first claim that for any R ′ ∈ R there is a constant M such that R iscontained in an M -neighbourhood of R ′ (and since R is finite we may then aswell assume that this constant M is independant of R ′ ). Indeed, because R isfinite, there is a finite index subgroup H ′ of H which stabilizes R ′ . In particulard( D, R ′ ) = d( H ′ · D, R ′ ) for any chamber D of R . Moreover, the chambers of R are contained in finitely H ′ -orbits since H acts transitively on R . The claimfollows. Let now J ′ ⊆ S \ J be minimal such that R := R J ∪ J ′ ( C ) contains the reunionof the residues of R . In other words, H < P J ∪ J ′ with J ′ minimal for this property.We next show that J ′ ⊆ J ⊥ . For this, it is sufficient to see that H stabilizes R J ∪ J ⊥ ( C ).Note that, given R ′ ∈ R , if A is an apartment containing some chamber C ′ of R ′ , then every chamber D in R ∩ A is at distance at most M from R ′ ∩ A . Indeed,if ρ = ρ A,C ′ is the retraction of ∆ onto A centered at C ′ , then for every D ′ ∈ R ′ such that d( D, D ′ ) ≤ M , the chamber ρ ( D ′ ) belongs to R ′ ∩ A and is at distanceat most M from D = ρ ( D ) since ρ is distance decreasing (see [Dav98, Lemma11.2]).Let now g ∈ H and set R ′ := gR ∈ R . Let Γ be a minimal gallery from C to itscombinatorial projection onto R ′ , which we denote by C ′ . Let A be an apartmentcontaining Γ. Finally, let w ∈ W = Stab G ( A ) / Fix G ( A ) such that wC = C ′ . Wewant to show that Γ is a J ⊥ -gallery, that is, w ∈ W J ⊥ .To this end, we first observe that, since Γ joins C to its projection onto R ′ ,it does not cross any wall of R ′ ∩ A . We claim that Γ does not cross any wallof R ∩ A either. Indeed, assume on the contrary that Γ crosses some wall m of R ∩ A . Then by Lemma 2.19 we would find a wall m ′ = m intersecting R ∩ A andparallel to m , and therefore also chambers of R ∩ A at unbounded distance from R ′ ∩ A , a contradiction.Thus every wall crossed by Γ separates R ∩ A from R ′ ∩ A . In particular, R ∩ A is contained in an M -neighbourhood of any such wall m since it is contained in an M -neighbourhood of R ′ ∩ A and since every minimal gallery between a chamberin R ∩ A and a chamber in R ′ ∩ A crosses m . Then, by Lemmas 2.1 and 2.20, thereflection associated to m belongs to W J ⊥ . Therefore w is a product of reflectionsthat belong to W J ⊥ , as desired.Finally, we show that J ′ is spherical. As R splits into a product of buildings R = R J × R J ′ , where R J := R J ( C ) and R J ′ := R J ′ ( C ), we get a homomorphism H → Aut( R J ) × Aut( R J ′ ). As O stabilizes R J and has finite index in H , theimage of H in Aut( R J ′ ) has finite orbits in R J ′ . In particular, by the Bruhat–Titsfixed point theorem, H fixes a point in the Davis realization of R J ′ , and thusstabilizes a spherical residue of R J ′ . But this residue must be the whole of R J ′ byminimality of J ′ . This concludes the proof of the lemma. (cid:3) Proof of Theorem 3.3.
The first statement summarizes Corollary 3.17 and Lem-mas 3.18 and 3.19, since some conjugate gOg − of O contains O as a finite indexsubgroup. The second statement then follows from Lemma 3.2 applied to theopen subgroup O of P J . Finally, the two last statements are a consequence ofLemma 3.19. Indeed, any subgroup H containing gOg − with finite index alsocontains O with finite index. Then H is a subgroup of some standard parabolic P J ∪ J ′ for some spherical subset J ′ ⊂ J ⊥ . Moreover, since the index of O in P J ∪ J ′ is finite, and since there are only finitely many spherical subsets of J ⊥ , it followsthat there are only finitely many possibilities for H . (cid:3) Remark 3.20.
Let O be a subgroup of G , and let J ⊆ S be as in the statementof Theorem 3.3. Assume that J ⊥ is spherical. Then L + J · U J ∪ J ⊥ has finite indexin P J ∪ J ⊥ and is thus open since it is closed. Thus, in that case, O is open if andonly if L + J · U J ∪ J ⊥ < gOg − < P J ∪ J ⊥ for some g ∈ G . PEN SUBGROUPS OF KAC–MOODY GROUPS 23
Corollary 3.21.
Let O be an open subgroup of G and let J ⊆ S be minimal suchthat O virtually stabilizes a J -residue. If J ⊥ = ∅ , then there exists some g ∈ G such that L + J · U J < gOg − < P J = H · L + J · U J . Proof . This readily follows from Theorem 3.3. (cid:3)
To prove Corollary B, we use the following general fact, which is well known inthe discrete case.
Lemma 3.22.
Let G be a locally compact group. Then G is Noetherian if andonly if every open subgroup is compactly generated. Proof . Assume that G is Noetherian and let O < G be open. Let U < O be thesubgroup generated by some compact identity neighbourhood V in O . If U = O ,there is some g ∈ O \ U and we let U = h U ∪ { g }i . Proceeding inductivelywe obtain an ascending chain of open subgroups U < U < · · · < O , and theascending chain condition ensures that O = U n for some n . In other words O isgenerated by the compact set V ∪ { g , . . . , g n } .Assume conversely that every open subgroup is compactly generated, and let U < U < . . . be an ascending chain of open subgroups. Then U = S n U n is anopen subgroup. Let C be a compact generating set for U . By compactness, theinclusion C ⊂ S n U n implies that C is contained in U n for some n since every U j is open. Thus U = h C i < U n , whence U = U n and G is Noetherian. (cid:3) Proof of Corollary B.
By Theorem A, every open subgroup of a complete Kac–Moody group G over a finite field is contained as a finite index subgroup in someparabolic subgroup. Notice that parabolic subgroups are compactly generated bythe Svarc–Milnor Lemma since they act properly and cocompactly on the residueof which they are the stabilizer. Since a cocompact subgroup of a group actingcocompactly on a space also acts cocompactly on that space, it follows for thesame reason that all open subgroups of G are compactly generated; hence G isNoetherian by Lemma 3.22. (cid:3) Proof of Corollary C.
Immediate from Theorem A since Coxeter groups of affineand compact hyperbolic type are precisely those Coxeter groups all of whoseproper parabolic subgroups are finite. (cid:3)
References [AB08] Peter Abramenko and Kenneth S. Brown,
Buildings , Graduate Texts in Mathematics,vol. 248, Springer, New York, 2008, Theory and applications. MR MR2439729[Bou68] N. Bourbaki, ´El´ements de math´ematique. Fasc. XXXIV. Groupes et alg`ebres de Lie.Chapitre IV: Groupes de Coxeter et syst`emes de Tits. Chapitre V: Groupes engendr´espar des r´eflexions. Chapitre VI: syst`emes de racines , Actualit´es Scientifiques et Indus-trielles, No. 1337, Hermann, Paris, 1968. MR 0240238 (39
Conjugacy of 2-spherical subgroups of Coxeter groups andparallel walls , Algebr. Geom. Topol. (2006), 1987–2029. MR 2263057 (2007j:20059)[Cap09] , Buildings with isolated subspaces and relatively hyperbolic Coxeter groups , In-nov. Incidence Geom. (2009), 15–31. MR 2665193[CF10] Pierre-Emmanuel Caprace and Koji Fujiwara, Rank-one isometries of buildings andquasi-morphisms of Kac-Moody groups , Geom. Funct. Anal. (2010), no. 5, 1296–1319. MR 2585575 (2011e:20061)[CM11] Pierre-Emmanuel Caprace and Nicolas Monod, Decomposing locally compact groupsinto simple pieces , Math. Proc. Cambridge Philos. Soc. (2011), no. 1, 97–128.MR 2739075 [CR09] Pierre-Emmanuel Caprace and Bertrand R´emy,
Simplicity and superrigidity of twinbuilding lattices , Invent. Math. (2009), no. 1, 169–221. MR 2485882 (2010d:20056)[Dav98] Michael W. Davis,
Buildings are
CAT(0), Geometry and cohomology in group theory(Durham, 1994), London Math. Soc. Lecture Note Ser., vol. 252, Cambridge Univ.Press, Cambridge, 1998, pp. 108–123. MR 1709955 (2000i:20068)[Deo82] Vinay V. Deodhar,
On the root system of a Coxeter group , Comm. Algebra (1982),no. 6, 611–630. MR 647210 (83j:20052a)[DJ99] A. Dranishnikov and T. Januszkiewicz, Every Coxeter group acts amenably on a com-pact space , Proceedings of the 1999 Topology and Dynamics Conference (Salt LakeCity, UT), vol. 24, 1999, pp. 135–141. MR 1802681 (2001k:20082)[H´ee93] J.-Y. H´ee,
Sur la torsion de Steinberg–Ree des groupes de Chevalley et des groupes deKac–Moody , Th`ese d’´Etat de l’Universit´e Paris 11 Orsay, 1993.[Kra09] Daan Krammer,
The conjugacy problem for Coxeter groups , Groups Geom. Dyn. (2009), no. 1, 71–171. MR 2466021 (2010d:20047)[NV02] Guennadi A. Noskov and `Ernest B. Vinberg, Strong Tits alternative for subgroups ofCoxeter groups , J. Lie Theory (2002), no. 1, 259–264. MR 1885045 (2002k:20072)[Pra82] Gopal Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a the-orem of Tits , Bull. Soc. Math. France (1982), no. 2, 197–202. MR 667750(83m:20064)[R´em02] Bertrand R´emy,
Groupes de Kac-Moody d´eploy´es et presque d´eploy´es , Ast´erisque(2002), no. 277, viii+348. MR 1909671 (2003d:20036)[R´em04] B. R´emy,
Topological simplicity, commensurator super-rigidity and non-linearities ofKac-Moody groups , Geom. Funct. Anal. (2004), no. 4, 810–852, With an appendixby P. Bonvin. MR 2084981 (2005g:22024)[RR06] Bertrand R´emy and Mark Ronan, Topological groups of Kac-Moody type, right-angledtwinnings and their lattices , Comment. Math. Helv. (2006), no. 1, 191–219.MR 2208804 (2007b:20063)[Tit74] Jacques Tits, Buildings of spherical type and finite BN-pairs , Lecture Notes in Math-ematics, Vol. 386, Springer-Verlag, Berlin, 1974. MR 0470099 (57
UCL, 1348 Louvain-la-Neuve, Belgium
E-mail address : [email protected] UCL, 1348 Louvain-la-Neuve, Belgium
E-mail address ::