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M. Burger, S. Mozes (2000)
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GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE by MARC BURGER and SnAnAR MOZES Introduction The group of automorphisms AutT of a locally finite tree T is a locally compact group. In this work we study a large class of groups of automorphisms of a locally finite tree which exhibit a rich structure theory, analogous to that of semisimple Lie groups. Recall that a rank one simple algebraic group G over a locally compact non-archimedean field acts on the associated Bruhat-Tits tree A. Thus G is a closed subgroup of Aut A. Moreover its action on A is locally oc-transitive, that is, the stabilizer of every vertex x acts transitively on all spheres of finite radius centered at x. In this paper we study the structure of closed non-discrete subgroups of AutT satisfying various local properties. Of particular interest will be the class of locally primitive groups, namely subgroups H < Aut T such that for every vertex x, its stabilizer H(x) acts as a primitive permutation group on the set of neighbouring edges. Given a totally disconnected locally compact group H, we define two topologi- cally characteristic subgroups: 9 H (~ which is the intersection of all normal cocompact subgroups of H, 9 QZ(H) which is the subgroup consisting of all elements with open centralizer in H. In our setting these groups will play a role analogous to the one played by the connected component of the identity and the kernel of the adjoint representation in Lie group theory. A corollary of the structure theory developed in this paper is that for any closed, non-discrete, locally 2-transitive group H < AutT, the group H(~)/QZ(H (~)) is topologically simple. In Chapter 1 we study the structure of closed non-discrete locally primitive groups of automorphisms of trees and more generally of graphs. The main results show that H (~/ is (minimal) cocompact in H whereas QZ(H) is a maximal discrete normal subgroup of H. Next we study the structure of H(~)/QZ(H (~)) and show that this group is the product of finitely many topologically simple groups. The structure theorem of locally primitive groups of automorphisms of a graph is complemented by a result (see Section 1.7) showing that such a graph may be obtained via a certain fiber product of graphs associated to the simple factors of H(~)/QZ(H(~)). Examples of 114 MARC BURGER, SHAHAR MOZES graphs and locally primitive groups of automorphisms may be obtained by considering actions of algebraic groups over nonarchimedean local fields on certain graphs "drawn" equivariantly on the associated Bruhat-Tits building, or more generally, the action of appropriate subgroups. Considering the action of a higher rank simple algebraic group G on subgraphs of the 1-skeleton of the corresponding Bruhat-Tits building leads to an action of an extension of G on a tree. In (1.8) a special case of this construction is analyzed in detail; namely we construct a "graph of diagonals" 12~ in the product of trees associated with the group L = PSL(2, Qp) x PSL(2, Qp). The semidirect product G of L with the automorphism switching the factors acts on !~" as a locally primitive group of automorphisms, and an extension H of G by the fundamental group nl(~) acts on the corresponding universal covering tree ~; we show then that QZ(H) coincides with ~1(~ r) and H (~ with the inverse image of L in H. In Chapter 2, the structure of small neighbourhoods of the identity (i.e. stabilizers of balls in T) in a locally primitive group H is studied. When the group H is discrete there has been a lot of interest in the structure and classification of such groups for example in connection with the Goldschmidt-Sims conjecture which asserts the fmiteness of the number of conjugacy classes of discrete locally primitive groups acting on a given tree. A basic result concerning the structure of discrete locally primitive groups is the Thompson-Wielandt Theorem which shows that the stabilizers of certain 2-balls in the tree are p-groups for some prime p. In general, when considering a non-discrete locally primitive group one cannot expect the stabilizer of some fixed size ball to be a pro-p-group. However, we obtain (Proposition 2.1.2) a substitute which holds also in the non-discrete case saying that when the stabilizer of a vertex contains a non trivial closed normal pro-p-subgroup, then already the stabilizer of a ball of radius 2 is a pro-p-group. In Chapter 3 we consider vertex transitive groups, in particular 2-transitive groups and show that under quite general conditions these are often either discrete or already o~-transitive; on the way, the results of Chapter 2 are used among other things to deduce the non-discreteness of certain subgroups of AutT. To every permutation group F < St on d elements we associate a closed subgroup U(F) < AutTd acting on the d regular tree Td. Every vertex transitive subgroup H < AutTd is conjugate to a subgroup of U(F), where F is permutation isomorphic to the permutation group induced by a vertex stabilizer in H on the neighbouring edges. We show that when F is a 2-transitive group, then U(F) is (x~-transitive; if moreover the stabilizer in F of an edge is simple the only closed non-discrete vertex transitive subgroup of AutT acting locally like F is U(F). These results go towards a classification of closed, non-discrete, 2-transitive subgroups of AutT. The study of discrete subgroups of AutT satisfying local transitivity properties can be traced back to the work of W. T. Tutte ([Tu]), who introduced the concept of "s-transitive graphs" and initiated their classification. Via the fundamental results of GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 115 Thompson and Wielandt alluded to above, this culminated in the classification of locally primitive discrete groups of automorphisms of the 3-regular tree by D. Goldschmidt ([Go]) and the classification of s-transitive groups of automorphisms of general regular trees, for s /> 4, by R. Weiss ([We]). From another viewpoint, the analogy between groups of automorphisms of a tree and Lie groups have led to various works exploiting the Bass-Serre theory to study general lattices in Aut T, see [Ba-Ku], [Ba-Lu], [Lu]. In the case of semisimple Lie groups, irreducible lattices in higher rank groups have a very rich structure theory and one encounters many deep and interesting phenomena such as (super)rigidity and arithmeticity. It turns out that in order to develop a structure theory for irreducible lattices in groups of the form AutTt x AutT2 one needs to consider lattices whose projections satisfy various "largeness" conditions such as being locally primitive. The results of the present paper are used in an essential way m [B-M]2 where we study the normal subgroup structure of irreducible cocompact lattices in AutT1 x AutT2 and in [B-M-Z] where the linear representation theory of such lattices is studied. Some of the results presented in this paper were announced in [B-M],. Acknowledgements. -- The second named author thanks the Forschungsinstitut far Mathematik (FIM) of ETH, Ztirich, for its hospitality, the Israel Academy Science Foundation for partial support and both authors thank IHES, Bures-sur-Yvette, where this work was completed, for its hospitality. Finally the authors want to express their gratitude to the referees for their numerous suggestions and remarks for improvement. O. Notations, terminology 0.1. A permutation group F < Sym(g~) of a set ~ is quasiprimitive if it is transitive and if every nontrivial normal subgroup e :~ N <1 F acts transitively on ~. Let F + = (F,, : 0) E ~) denote the normal subgroup of F generated by the stabilizers F0~ of points r E ~. We have the following implications: F is 2-transitive =~ F is primitive ~ F is quasiprimitive F=F or F is simple and regular (that is simply transitive) on ~. Recall that a permutation group F < Sym~ is called primitive if it is transitive and if every F-invariant partition of f~ is either the partition into points or the trivial partition {~}. An equivalent condition which is often used in the sequel is that F is transitive and the stabilizer F~ of a point m E ~ is a maximal subgroup of F. See [Di-Mo] Chapt. 4 for the structure of primitive and [Pr] w for the structure of quasiprimitive groups. 116 MARC BURGER, SHAHAR MOZES 0.2. For notations and notions pertaining to graph theory we adopt the viewpoint of Serre ([Se]). Let I~=(X,Y) be a graph with vertex set X and edge set Y, let E(x)={y G Y: o(y)=x} denote the set of edges with origin x; for a subgroup H < Aut IJ let H(x) = StabH(x ) and H(x) < Sym(E(x) ) be the permutation group obtained by restricting to E(x) the action of H(x) on Y. We say that H is locally "P" if for every x E X, the permutation group H(x) < Sym(E(x)) satisfies one of the following properties "P": transitive, quasiprimitive, primitive, 2-transitive. We say that H is locally n-transitive (n) 3) if, for every x E X, the group H(x) acts transitively on the set of reduced paths (i.e. without back-tracking) of length n and origin x. Observe that H is locally 2-transitive iff, for every x 6 X, H(x) acts transitively on the set of reduced paths of length 2 and origin x. We say that H < Aut 9 is n-transitive, n/> 1, if H acts transitively on the set of oriented paths of length n without back-tracking; H < Aut IJ is locally oc-transitive if it is locally n-transitive for all n ) 1. For a connected graph 9 and H < Autg, we have H < Sym(Y), and H + denotes the subgroup generated by edge stabilizers; +H denotes the subgroup generated by all vertex-stabilizers. If 9 = (X, Y) is connected and locally finite, the group Aut 9 < Sym(Y) is locally compact for the topology of pointwise convergence on Y. Let d denote the combinatorial distance on X, n/> 1 and Xl, ...,xk 6 X; g(y)=y for ally E X ] H,(xl, ..., xk) = /g E H" with d(y, {x,, ...,x~} ) <<. n and for x 6 X, we set H.(x) = Hdx)lH,+~ (x). For x,y G X adjacent vertices, set H(x,y):= H(x) f"l H(y). 1. The structure of locally primitive groups 1.1. Let H be a locally compact, totally disconnected group. Define H(~) := N L L<H where the intersection is taken over all open subgroups L < H of finite index, and QZ(H) = {h E H: Zn(h) is open}. Then H (~) and QZ(H) are topologically characteristic subgroups of H, and H (~ is closed. These definitions are motivated by the following: Example 1.1.1. -- Let G--G(Qp), where (3 is a semisimple algebraic group defined over Qp. Then G (~) coincides with the subgroup G(Qp) + generated by all GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 117 unipotent elements in G(Qp), and QZ(G) coincides with the kernel of the adjoint representation of the p-adic Lie group G. The first statement follows from [Till (main theorem) and [Bo-Ti] 6.14, while the second follows from the fact that Ad(g), g E G, is the identity if and only if g centralizes a small neighbourhood of the identity e C G. As any compact totally disconnected group K is profinite (Cor. 1.2.4 [Wil]), we have K (~) = (e) and hence H (~) = 71 N, where the intersection is taken over all N<3H closed, cocompact normal subgroups of H. Thus, every normal cocompact subgroup of H contains H(~); at the other extreme, every discrete normal subgroup of H is contained in QZ(H). While H (~) is closed, this need not be so for QZ(H), as is seen in the following: Example 1.1.2. --Let K=F N, where F is finite, centerless. Then QZ(K) coincides with the direct sum | F, which is a countable and dense subgroup of K. Let 0 be a locally finite graph and H < Auto a closed subgroup; then H is locally compact totally disconnected and the most basic issue concerning its structure is the control over closed normal subgroups of H. In this problem, H (~) and QZ(H) are relevant objects since they control cocompact normal, resp. discrete normal subgroups. However, at this level of generality not much can be said about the size of H (~176 and The general theme of this chapter is that in requiring local transitivity properties of H < Aut O, one obtains a good control over normal closed subgroups of H, and ends up with a class of locally compact groups behaving in many respects like semisimple Lie groups over local fields. In Sections 1.1 to 1.5 we develop the basic structure theory of locally quasiprim- itive groups. Our main goal is then the decomposition theorem (Thm. 1.7.1, Cor. 1.7.3) which describes how locally primitive groups are built up from topologically simple pieces; this result is reminiscent of the O'Nan-Scott theorem about (fmite) primitive permutation groups. 1.2. In this section O = (X, Y) denotes a locally finite connected graph with vertex set X and edge set Y. Proposition 1.2.1. -- Let H < Auto be a closed subgroup. We assume that H is non-discrete and locally quasiprimitive. 1) H/H (~ is compact. 2) QZ(H) acts free~ on X; it is a discrete non cocompact subgroup of H. 3) For any closed normal subgroup N <~ H, either N is non-discrete cocompact and N > H (~) or N is discrete and N C QZ(H). Moreovo;, the subgroup H (~) enjoys the foUowing properties: 4) QZ(H (~)) acts fre@ without inversions on O; more precise~, QZ(H (~)) = QZ(H)7/H (~). 118 MARC BURGER, SHAHAR MOZES 5) For any open normal subgroup N <1 H (~), we have N : H (~). 6) H (~) is topologically perfect, that is H (~) = [H(~), H(~) ]. In other words, there exists a unique minimal normal cocompact subgroup H (~ and a unique maximal normal discrete subgroup QZ(H); and every closed normal subgroup of H is either cocompact or discrete. As we shall see in the course of the proof, H (~) admits an edge or a complete star as precise fundamental domain. However, H (~176 needs not be locally quasiprimitive, thus 4) and 5) are not formal consequences of 1) and 2). Moreover, the group H(~)/QZ(H (~)) usually fails to be topologically simple; informations about its normal subgroups will be obtained in Section 1.5. In order to illustrate the objects occuring in Proposition 1.2.1, we present the following: Example 1.2.1. -- Let Ap be the Bruhat-Tits building associated to PSL(3, Qp), 13p the subgraph of the 1-skeleton of Ap consisting of all edges of a fLxed given label and T the universal covering tree of 1~, which is regular of degree #P2(FJ =p2 +p + 1 (the number of points in the projective plane over a field ofp elements). Let 1 , nl(IJJ , H~ , PSL(3, Qp) , 1 be the exact sequence associated to the universal covering projection T --~ gp. Then PSL(3, Qp) < Aut~lp and Hp < AutT are non-discrete, locally 2-transitive (since the linear group acts 2-transitively on the points of the projective plane); in particular, Proposition 1.2.1 applies. Notice that there is a natural map from PSL(3, Zp) (the stabilizer of a fixed base vertex of the building z~) into Hp so that its composition with the projection Hp --% PSL(3, Qp) is the identity. Let us denote the image of PSL(3, Zp) by Kp < Hp. It turns out that (a) H~) = Hp (b) QZ(Hp) = n1(gJ and the extension Hp of PSL(3, Qp) has the following algebraic connectedness property: (c) [Hp, nl(g~)] =nl(gJ, in particular H~ is perfect. Let us briefly outline the argument for showing (a)-(c), see also 1.8.1 for similar considerations. Let N <1 Hp be a finite index closed normal subgroup, let A = N Yl nl(gJ. Since A is of finite index in nl (gJ there is some g G A represented by a loop in a fixed apartment ~/~ of @, and such that the intersection of this loop with a certain half apartment ~-~+ consists of 3 edges forming half a basic hexagon (recall that the graph l~p can be viewed as consisting of small hexagons glued together). Now one can verify that the commutator [h, g] ofg with an element h 6 Kp whose action on the apartment Jg fixes the complement of ./~+ and moves ~g+, is an element of A < nl(fl~) which up to conjugacy is represented by a closed basic hexagon. It follows now easil~ using the transitivity of the action of Hf, that [Hp, A] =n~(13J. This in particular proves (c). GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 119 It follows also that N D ~l(~]p) and since Hp/n~(gp)=PSL(3, Qp) is simple it follows that N must be all of Hp and (a) is proved. To see (b) observe first that any element g 6 nl(gJ commutes with the open neighbourhood of the identity consisting of elements stabilizing the lift in T of a path representing g. Hence rcl(gp) C QZ(Hp). The reverse inequality follows easily by considering the structure of the action of elements in PSL(3, Qp) on the building Ap. Example 1.2.2. (see 3.2).-- Let F < Sd be a finite permutation group on d letters and U(F) < AutTd the associated universal group; then at every vertex x of Td, U(F)(x) < SymE(x) is permutation isomorphic to F < Sd; if F is transitive and generated by stabilizers, then U(F) + is of index 2 in U(F) and simple. One has then: U(F)/~) = U(F) + and QZ(U(F))= (e). This applies for instance to F < Sd, quasiprimitive, non-regular. Example 1.2.3. -- For the class of locally primitive groups obtained via the graph of diagonals, we refer to 1.8. Example 1.2.4. -- The group H=PGL(2, F2((x))), considered as a closed subgroup of the group of automorphisms AutT3 of the 3-regular tree, is locally primitive (in fact oo-transitive). We have H (~/= PSL(2, F2((x))); since H/H (~ _~ F2( (x) )*/Fz( (X) ) .2 , this gives an example where H (~) is of infinite index in H. We turn now to some corollaries. As an immediate consequence of Proposi- tion 1.2.1: 1), 5), 6), we obtain: Corollary 1.2.2. -- Let H < Autl~ be as in Proposition 1.2.1 and G < H a closed subgroup containing H (~). 1) G/[G, G] is compact. 2) For any open normal subgroup N <1 G, one has N D H (~). In particular, G (~176 = H (~176 For the next corollar~ observe that if H < Autg is locally quasiprimitive, then +H is of index at most two in H (see 1.3.0), and locally quasiprimitive as well since H(x) = +H(x), V x r X. Corollary 1.2.3. -- Let H < Autg be as in Proposition 1.2.1. 1) /fH > +H, and x,y are adjacent vertices, we have +H=H(x,y).H I~l. 2) /f H = +H, there exists v 6 X, with +H = H(v).H(~ 120 MARC BURGER, SHAHAR MOZES 1.3. In this section we collect a few general facts used in the proof of Proposition 1.2.1. 1.3.0. Let g=(X, Y) be a connected graph, H < Autg be a locally transitive subgroup. Then +H coincides with the kernel of the homomorphism ~ : H ~ Z/2Z, )~(g) = d(gx, x) mod 2. The group +H is transitive on the set of geometric edges and thus equals H(x) :r 1.3.1. Let g =(X, Y) be a connected graph, H < Autl~, g'= (X', Y') a connected subgraph of g and R C H such that, for all x' E X' and y E E(x'), there is r E R with ry E Y'; then A := (R) < H satisfies tO ~,g' = g. )~EA Fact 1.3.1 implies 1.3.2. Let g = (X, Y) be connected, locally finite and H < Aut g with H\t~ finite. Then there exists a finitely generated subgroup A C H with A\g finite. 1.3.3. Let t~=(X, Y) be connected, locally finite and A < Autg with A\g finite. Then ZAutt~(A) is a discrete subgroup of Autl~. Indeed, let B C Y be a finite subset with tO ~,B = Y and U < ZAut~(A) an open ~,cA subgroup with u(b)= b, V u E U, V b C B. Since U commutes with A, it acts like the identity on tO )~B =Y, which implies that U= (e) and that ZAut~(A) is discrete. ~cA 1.3.4. Let t~ = (X, Y) be connected, locally finite; let A1, A2 be subgroups of Aut g such that Al \g is finite and [Al, A2] < Aut g is discrete. Then A2 is discrete. Indeed, pick R C A1 finite such that (see 1.3.2) (R)\g is finite; take U C A2 open such that [r, U] =e, Vr E R. Thus U C ZAut~((R)) , and hence U is discrete by (1.3.3). This implies that A2 is discrete. 1.3.5. Let g=(X,Y) be connected, locally fmite, and G < Autg non-discrete. Then, QZ(G)\t~ is not fmite. Indeed assume that QZ(G)\I~ is finite and pick (using 1.3.2) a finitely generated subgroup A < QZ(G) with A\g finite. Since A is fmitely generated there is U < G open with U < ZA~t~(A) which by 1.3.3 implies that U, and hence G, is discrete; a contradiction. 1.3.6. Let g=(X, Y) be locally finite, connected, and F < Autg, discrete with F\g finite. Then NAut.(F) is a discrete subgroup of Aut~. Indeed, apply 1.3.4 with Al =F and A2 =NAut.(F). 1.4. In this section t~ = (X, Y) is a locally finite, connected graph. Let A, H be closed subgroups of Aut 1~ with A <~ H. We define, A < N <~ H, N is closed J//~ (H, A)= {N <H: and does not act freely on X GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 121 The set .//~f (H, A) is ordered by inclusion; let ~t/gnf (H, A) be the set of minimal elements. Lemma 1.4.1. -- Assume moreover that H\I~ is finite and that H does not active@ on X. Then (H, A) Proof. -- We use Zorn's Lemma. Observe that JV~,f (H, A)~: 0. Let 4 C M/~,f (H, A) be a chain and F a finite set of representatives of H\X. For every N E 4, the set FN={fE F : _N(f) < SymE(f) is not trivial } is non void. Since F is finite and 4 is a chain it follows that 7/N~FN is non void. Thus there existsfE F such that for every N E 4, N(f) is nontrivial which implies that M:= f-lN~ N(f) is not trivial. For gE M,g~:e, and N E 4, the set Ng: - {n E N(f)'nlE(f)=g } is a compact non-void subset of H(f) and, since W is a chain any finite subset of {N g : N E 4} has nonempty intersection. Thus 71 Ng ~: 0, and therefore N~- := 71 N does not act NE~ NEW " freely on X which shows that 4 admits a lower bound in .A/~/(H, A). [] We turn now to locally quasiprimitive subgroups of Autg. Lemma 1.4.2. -- Let H < Autg be a local~ quasiprimitive subgroup, N <~ H a normal subgroup and Xl(N) := {x E X: _N(x) acts transitive~ on E(x)} X2(N) := {x E X: N(x)= e}. One of the following holds: 1) X2(N)= X and N acts free~ on X. 2) Xl(N) = X and N acts transitiv@ on the set (geometric edges of 1~. 3) X=XI(N)I IX2(N) gives an H-invariant 2-colouring of g; for any x~ E Xg(N), the 1-neighbourhood ~/~x2, 1) is a precise fundamental domain for the N-action on 1~. Proof. --The subsets XI(N),X2(N) are H-invariant and, since H is locally quasiprimitive, X=XI(N) t_J Xz(N). If N does not act freely on X, there exists z E X with N(z) ~: {e} and, since g is connected, there must exist an N(z)-fixed vertex x E X, with N(x)~: (e). Then N(x) acts transitively on E(x) and XI(N)~: 0. Either X2(N)= 0, N is locally transitive and we are in Case 2), or X2(N)~: 0. Since H acts transitively on the set of geometric edges (see 1.3.0) it has at most 2 orbits in X. Since both XI(N), X2(N) are non void and H-invariant, they are exactly these 2 orbits. Since any pair of adjacent vertices {xl, x2} is a fundamental domain for the H-action on X, we conclude that if x2 E X2(N) then Xl E Xl(N). Thus every terminal vertex of ~x2, 1) is in XI(N) and we are in case 3 by 1.3.1. [] 122 MARC BURGER, SHAHAR MOZES Lemmas 1.4.1 and 1.4.2 play an important role in our study of the structure of locally quasiprimitive groups, and in the context of Proposition 1.2.1 we shall use them to identify H (~176 with the smallest closed normal subgroup of H whose action on X is not free. Proof of Proposition 1.2.1 (1) Let N <1 H be closed, cocompact; since H is non-discrete, N is non-discrete (see 1.3.6) and hence N E ~/~,$ (H, e). Conversely, if N E JVs (H, e) then N is cocompact in H, by Lemma 1.4.2. Thus H (~) = n N, where the intersection is over all N's in ~/~,f (H, e). Take M E J~,y(H, e) (Lemma 1.4.1) and N E JVs e); assume that N;bM. Then N N M D [N, M] is discrete, hence N, M are discrete (see 1.3.4) and therefore H C JVAut~(N ) is discrete (see 1.3.6), a contradiction. This shows that H (~/=M E J~,y(H, e) and proves 1). (2) Follows from Lemma 1.4.2 and 1.3.5. (3) Let N <~ H be a closed normal subgroup; either N acts freely on X, in particular N is discrete and hence contained in QZ(H), or N does not act freely on X, thus it is cocompact in H (Lemma 1.4.2) and hence contains H (~). (4) The inclusion QZ(H)n H (~176 C QZ(H (~176 follows from the definitions. The group QZ(H(~ being topologically characteristic in H (~176 is normal in H; if QZ(H (~)) (~ QZ(H), then QZ(H (~176 is not discrete, hence acts non-freely on X and QZ(H(~))\g is finite (Lemma 1.4.2); but (1.3.5) implies then that H(~)\g is not finite, contradicting 1). Thus QZ(H (~)) is a discrete normal subgroup of H and therefore contained in QZ(H). This proves 4). (5) Since H (~176 is cocompact in Aut g, and non discrete, J~,f (H (~176 e) ~= {a (Lemma 1.4.1); since QZ(H (~176 acts freely on X, every N E ~f (H (~ e) is non-discrete. Given O <a H (~) open and N E ~r (~ e), the group ONN is non-discrete normal in H (~176 in particular O n N acts non-freely on X hence O n N--N. Thus O contains the closed subgroup of H (~~ generated by the elements of ~/~,f (H (~176 e); this latter group being closed, normal in H and non-discrete, we conclude O = H (~). (6) Proposition 1.2.1, 1) and 1.3.4 imply that [H (~/, H (~)] is not discrete, thus [H(~), H( ~~ is non-discrete, closed and normal in H which implies [HIll, H(~)] = H/~/ by 3), above. [] Proof of Corollary 1.2.3. -- Assume first that g=T is a tree. Let U, <~ H be a decreasing sequence of open finite index subgroups with H (~/ C Un C +H GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 123 and Yl Un=H(~); let Fn=+H/Un, 7rn : +H ---+ F, the canonical projection and n)l D, := n,(H(x)) *~,(H(x,y))n,(H(y)) which we endow with the discrete topology. Since +H= H(x)*n(x,y)H(.y) (see 1.3.0), the universal property of amalgams implies that n, t +H is the composition of a continuous homomorphism 7t, : --. D, and the natural quotient map D, --- F,. Since D, is discrete, Kern', <~ +H is open, non-discrete, hence Kerff~ D H (~), which implies that Dn is compact and hence finite. This implies that rc,(H(x,y)) equals either ~,(H(x)) or ~,(H(y)) and hence rc,(H(y)) or ~,(H(x)) equals F,. If H > +H, the group H acts transitively on X and H(x), H(y) are H-conjugate; it follows then from the equivariance of 7r, w.r.t, conjugation by H, that nn(H(x,y))= F,. Since this holds for all n/> 1 and H(x,y) is compact, we obtain +H = H(x,y)-H (~ In the former case, there is a vertex v E X and a sequence nk ---* cx~, for which n,k(H(v))= F,,, which implies +H = H(v).H (~176 In general, let p : T ~ 9 be the universal covering of t~ and 1 ~ ~i(9) ~ G ~ H ~ 1 the associated exact sequence. The corollary follows then from the facts: ~- I(H(~)) D G (~ , ~(G(x) ) = H(p(x) ), V x E Vert T. [] 1.5. In this section we turn our attention to normal subgroups of H (~). Our main result is: Proposition 1.5.1. -- Let H < Autg be a closed subgroup. Assume that H is non-discrete, locally quasiprimitive. Let A <~ H, with A C QZ(H(~/). 1) a) H acts transitiv@ on JPg,y (H/~ A). b) The set Jtg,f (H (~ A) is finite. 2) Let M E ,/~.f (H (~176 A); a) M/A /s topologically perfect. b) QZ(M) actsfieely on X; QZ(M) = QZ(H (~)) N M. c) M / QZ(M) is topologically simple. 3) For every N E ,/Unf (H (~), A) there is M E ,/~,f (H (~), A) with N D M. Corollary 1.5.2. -- Minimal normal closed nontnvial subgroups of H(~)/QZ(H (~)) exist; they are all H-conjugate, finite in number and topologically simple. Proof of Proposition 1.5.1. -- Since every discrete normal subgroup of H (~) is contained in QZ(H (~)) and the latter acts freely on X (Proposition 1.2.1, 4)), it follows that every element of J//~f (H (~), A) is non-discrete; this and 1.3.4 implies (1) For every N E J//~f (H (~ A), we have [t-I (~ N] ~ QZ(H(~)). For ~" C ~,j(H (~ A), let M~ := (M 9 M E ~), that is the subgroup of H (~ generated by U M. MEg (2) H acts transitively on ,/Zg,f (H (~ A). Otherwise, let ~ C JC~,j(H (~), A) be an H-orbit and M ~ ~.. For every N E ~,, the subgroup [N, M] C N V/M acts freely on X, is discrete and normal in H (~ 124 MARC BURGER, SHAHAR MOZES Hence [N, M] C QZ(H(~/), which implies [M~-, M] C QZ(H(~/). On the other hand, since ~ is an H-orbit, M~ is closed, normal in H and non-discrete, hence (Prop. 1.2.1, 3)) M~ =H/~/. We conclude [H(~/, M] C QZ(H(~ which contradicts (1). (3) We have [M, M] .A = M, V M E JFgf (H/~/, A). Otherwise, there exists M0 C JEg,f (H/~/, A) with [M0, M0].A < M0, hence [M0, M0].A acts freely on X, is discrete and thus [M0, M0] C QZ(H(~/); (2) implies then [M, M] C QZ(H(~/), VM E ~/Pg,f (H (~t, A). Since on the other hand [M, M'] C QZ(H (~)) for all M =~ M' in JEg,j(H (~) , A), one concludes easily that [H (~), H/M)] C QZ(H(~)), contradicting (1). (4) For every N C A~,f (H/~) , A), there is M E ./~g,/H (~/, A) with N D M. Indeed, let ~ = {M E ./~.j(H (~ A) : N ;b M}. Then we have [Ma, N] C QZ(H(~ on the other hand, for .~r=./~nj(H(O~), A), the group H~r C H (~ is closed, non-discrete and normal in H, thus M~=H (~ Using (1) we conclude that ~ ~:./~nf (H (~ A), which establishes (4). (5) Let ~, ~' be disjoint subsets of ./~.f (H (~ , A). Then Ma Cl M~ C QZ(H(~176 Indeed, otherwise we have H a N H a, E .A~.f (H (~), A) and there exists (by (4)) M E "~nf (H(~ A) with M C H~ Yl Mg,,. But this implies [M, M] C [M~, H~.,] C QZ(H(~ which contradicts (3). (6) ~/~f (U (~ A) is finite. Let G = tO M~., where the union is over all fmite subsets ~ C J~u (H (~ A); then G is non-discrete, normal in H hence G = H (~ Since H (~) is a separable metric space, the same holds for G and therefore there exists a countable dense subgroup L C G. Fix an exhaustion Fn C F~+I C ... C L of L by finite subsets and let ~ C N'~+I, [~.1 < +e~, be such that Fn C M~., for all n /> 1. Thus L C M~ and hence M~ =H (~), which by (5) and n=l n=l (1) implies ~/~,f (H (~) A)= ~ ~', and hence ~g/g,f (H (~) A) is countable. Fix ' n=l M E ~/~,f (H (~/, A); the closed subgroup NH(M) is then of countable index in H, hence has non-void interior. Thus NH(M) is open in H, contains H (~), and hence is of finite index in H. Since H acts transitively on ~/~nf (n(cx~) , J~) this implies that the latter is finite. So far we have proved assertions 1 a), b), 2 a), and 3) of Proposition 1.5.1. We turn now to the remaining assertions. 2 b) Let "/~nf (H(~ , A) = {M1, ..., Mr} (see 1 b)) and define a = QZ(M1) 9 ... 9 QZ(Mr). GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 125 Then ~ is a normal subgroup of H; if f~ does not act freely on X, ~\g is finite (Lemma 1.4.2), and (1.3.2) there exist El, ..., ~k in f~ such that for ~' := @~, ..., )~k), fl'\g is finite. Let ~.i = ai'bi, ai E QZ(MI), bi E QZ(M2) 9 ... 9 QZ(Mr), 1 ~< i ~< k; let U1 < MI be an open subgroup with [ai, U1]=e, 1 <~ i <~ k; since [M2- ...- Mr, M1] C QZ(H(~)), there exists an open subgroup U2 < M1 such that [bi, U2] = e, 1 ~< i ~< k. The open subgroup U := U1 M U2 < MI is therefore contained in Zautg(~'~ t) which implies (1.3.3) that U and hence M1 is discrete, a contradiction. This shows that fl acts freely on X, is discrete and hence C QZ(H(~)), that is, QZ(Mi) C QZ(H (~/) 71 Mi; the opposite inclusion follows from the definitions. This shows 2 b). 2c) Let M E J~,f (H (~/, A) and N <1 M a closed subgroup with N D QZ(M). For any M' E "~/~nf ( u(~/, A) with M' :~ M, we have [M', M] C M' 71 M C QZ(H(~/), which implies that [M', N] C QZ(H (~/) 71 M = QZ(M) C N; thus M' normalizes N. Since N <1 M, this implies that N <~ H (~/ and hence, by minimality of M, we have either N = M or N acts freely on X and N C QZ(H (~/) M M = QZ(M). [] 1.6. Fiber products 1.6.1. In the theory of fmite permutation groups, wreath products are used to build new primitive actions out of old ones; fiber products of graphs, whose basic properties we look at now, play a similar role in the theory of locally primitive groups. Let q0 i : ~i ~ ~, 1 ~< i ~< r, be surjective morphisms of graphs, where gi-- (Xi, Yi), [) = (V, E); the fibered product of the graphs gi relative to the morphisms ~0 i is the graph YI,i gi = (X, Y) where { r } X = yI~iXi = (xl, ..., Xr) E Hi= IXi : qJi(Xi) = qJj(X/), V i,j Y= 1-LiYi = {(yl,...,Y~) E I-[~=lYi : q~i(Yi)=g~j(yj), Vi,j} and the origin, terminus maps are given by o((yl,'..,yr)) =(O(yl),'",O(yr)), l((yl,...,yr))-=~ (/(Yl), ..., /(Yr) ) 9 This graph p = 1-I, i l~i comes with projection morphisms Pi : P ---+ l~i and a product morphism q0 : p ---+ [), such that the diagram Iti commutes for all i, 1 ~< i ~ r. This latter property could be used to give a categorial definition of fibered products; we leave this point to the reader. 126 MARC BURGER, SHAHAR MOZES Let Aut,ig i be the group of those automorphisms of gi which permute the fibers of q~i, and ni : Aut,il~i ~ Aut ~ the corresponding homomorphism, with respect to which q0i is equivariant. The fiber product G= 1-I~iAut,ig i is a subgroup of Autp; it comes with projection homomorphisms Oi : G --+ Autgi with respect to which Pi : P ~ l~i is equivariant, a product homomorphism x : G ~ Aut[?, w.r.t, which q0 : p ---+ [~ is equivariant and such that ~ = xi o Of, for all 1 ~< i ~< r. When gl .... =gr=It, q0Z ..... q0r=q0, which is our main case of interest, we denote by 1-I~ ) 1~ the r-fold fiber product of g relative to q0, I-[(r~ q) (r), 9 I-I o 1~ ---+ [1 the product morphism, n : Aut,pg ---+ Aut [? the natural homomorphism, 1-I(~ / Aut ~ol~ < Aut (I-[~) 1~) the r-fold fiber product of the group Aut,l~ relative to ~, and finally, I-Fin : I-[~ ) Aut t,l~ Aut [~ the associated product homomorphism. The permutation of factors realizes the symmetric group S~ on r letters as subgroup of Aut (1-I~) g); given any subgroup H < Aut,l~, I-[(~ ) H is a subgroup of I-[~ ) Aut,l~, normalized by S, We turn now to the following special situation: IJ=(X, Y) is a locally finite, connected graph, H < Autl~ is closed, non-discrete, locally primitive, M <~ H is a closed normal subgroup which acts non-freely on X and without inversions on g; let q--M\I~ be the quotient graph and rc : Aut,t~ ~ Aut q the canonical homomorphism. Observe that H C Aut,g, let r t> 1 and W C Sr be a transitive subgroup. Proposition 1.6.1. -- The semidirect product (II:) ) [Zl--T(r)'~ G= H x W < aut ~ll~og) is closed, non-discrete; it is local~ primitive if and on~ /f, _H(x) < Sym E(x) is non regular whenever _M(x) < Sym E(x) is non-trivial. In this case, G (~) = M (~) x ... x M (~) QZ(G) = QZ(H) x~ ... x~ QZ(H) = x ... x Remark 1.6.1 9 --The main point of Proposition 1.6.1 is the one concerning (local) primitivity; the other statements are formal consequences of the definitions and left to the reader. Remark 1.6.2. -- If in the above context g is a tree and M = H (~), then (see Corollary 1.7.7) H(x) < SymE(x) is non-regular whenever _M(x) < SymE(x) is non- trivial, and the conclusions of Proposition 1.6.1 hold. Using 1.4.2 we compute now the local permutation groups of G; let 1-[ (r) = (v, E); we distinguish two cases: GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 127 1) The quotient q=M\l~ is an edge; thus +H C Kern, and for any vertex v = (x, ..., x) of diagonal type, the group _G(v) < Sym E(v) is permutation isomorphic to the wreath product H__(x)r~ W on E(x) r, the latter is primitive if and only if W is transitive and H_(x) < Sym E(x) is primitive, non-regular (see [Di-Mo] or Lemma 1.6.2 below). Finallg observe that any vertex of gI ,~ (r) t~ is G-conjugate to a vertex of diagonal type. 2) The quotient q = M\t~ is a star. Let v= (x, ..., x) be a vertex of 1-I~ ) g, such that M(x) acts regularly on E(x); then _G(v) < SymE(v) is permutation isomorphic to H(x) < Sym Let v= (Z, ..., z) such that _M(z) acts transitively on E(Z); let ~: H_(Z) ---* g_(z)/M_(z) be the canonical projection; then _G(v) < SymE(v) is permutation isomorphic to the action of \/(H-(z)x~-"'x~-H-(z)~ >4 W on E(z)r; the Proposition 1.6.1 follows then from Lemma 1.6.2. -- Let F < Sym~'l, I~1 /> 2, r/> 2, and W < Sr befinite transitive permutation groups; /et N <3 F, D(F) < F r the diagonal subgroup and a = D(F).N r >4 W. Then, the G-action on ftr is primitive if and on# if No, 89 (e) , co Eft, and the F-action on ft is primitive. Proof. -- If G < Sym (f~') is primitive, then F r >4 W is primitive and hence F < Sym a is primitive; for x--(co, ..., co) we have Gx = D(FO,).N~, >~ W; if No, = (e) and N ~: e, then L := D(F) >4 W satisfies, Gx < L < G and G is not primitive; if N = e, then G is not transitive. This shows the necessity of the above conditions. Conversely, let L < G with Gx < L < G. Let N (') be the ith factor of N r viewed as subgroup of N r and Pi : Fr ~ F, the projection on the ith factor. We have N(~ C L n N (*~ <3 L n N r, and thus: No, C pi(L n N ('~) <1 pi(L n N ~) C N. Since D(F~0) normalizes L n N (') and L n N r, Fo, normalizes pi(L n N (*~) <1 pi(L n Nr). We observe now that if No, < U < N, and F~o normalizes U, then either U = No, or U = N; indeed F~o.U(> Fo,) is a subgroup of F and Fo, is maximal in F. We have thus two cases: 1) pi(L N N (*)) = N for one, and hence by transitivity of W, every i. Thus L D N r, hence D(FO,).Nr = D(F~o.N)-Nr = D(F).N r is contained in L which implies L--G. 2) pi(LnN(*?) = N,o, for one and hence all ,'s. Ifpi(LNNr) =N, we get No, <3 N and hence No, = (e) since N acts transitively. This is a contradiction; thus pi(L N N r) = No, for all z's and hence L n N r= (No0) r. Since D(F).N r= D(FO,).N r, we have L n F= D(FO,).(L n N r) = D(Fc0)(Nc0) r, and hence L = Gx. [] 1.6.2. Let I~=(X, Y) be a graph and G < Autt~. In this subsection we will show that when G is a direct product, then under certain conditions, g is in a natural way 1213 MARl2 BURGER, bttAHAR MOZES a fiber product. More precisely, assume that there are normal subgroups M1, ..., M, in G such that (i) Mifl Ms. = (e), V i ~:j. (ii) G=M1....-M,. (ui) G(x) Mr(x), Vx E X. Let M~ be the product of all factors M1, ..., Mr except Mi, 9i = M~\I~, qi: 1~ ---+ ~ti, Pi : l~i ~ Hi\l}i = G\9, where Hi is the image of G in Aut gi and q:l}~gl xp 1...x#~g, defined by q(x)=(ql(x), ...,qr(X) ) , X E X, q0') = (ql0'), ..., qr(Y)), y E Y. Proposition 1.6.3. -- The morphism of graphs q. ---4 ~PH'i~ i 9 r is an isomorphism; it is equivariant with respect to the isomorphism G ---+ 1-Ii= 1Hi 9 Proof. -- This follows immediately from the analogous statement where G < Sym X is a permutation group of a set X; thus let Xi = M~\X, A= G\X, qi : X ) Xi , Pi : Xi ) A, and q(x) = (ql (x), ..., qr(X) ). 1) q is injective: Let x,y E X with qi(x)= qi(Y), V i, that is M~x=M~y. For every i, we have then y m~ 0 (z~ .._(0 , (,) = "....m r x, where % E Mj and Trt i = e. It follows then that m m ... m~ s) m x=x, Vi,j, which by (iii) implies (/ / (m~S))-' m~) E Me(x), and hence for g=J, "5-(') E Mj(x), for all i,j which ilnplies y = x. q is surjective: Let xi E Xi, with pi(xi)=pj(xj); pick Yi E X with xi = M~Yi; observe 2) that G~i= G~ V i,j and thus writing G~i= G~, we find mi E Mi such that xi=Mlimi ~. Define now hi =ml "..." mi'... "mr, where mi is omitted in the product; then hi E MI, him[s = hjmfs V i,j and thus defining x = him[s we get qi(x) = xi, for all i. [] GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 129 1.7. The structure of local~ primitive groups The results of 1.5 and 1.6 suggest that a locally primitive group G < Autg is built out of several copies of a locally primitive "almost-simple" group M < AutTJI, and that the graph 9 is related to an appropriate fiber product of copies of 9)I. The purpose of this section is to show that this is indeed the case when g is a tree. 1.7.1. Let T=(X, Y) be a locally finite tree, H < AutT a closed, non-discrete, locally quasiprimitive group and A < QZ(H (~)) with A <3 H. Then we know (see 1.5) that the set J~nf (H(~), A) of minimal closed non-discrete normal subgroups of H (~) containing A is non-void, finite and that H acts transitively on it by conjugation. We have Theorem 1.7.1. -- Assume moreover that H < AutT is locally primitive; let ._.~f (n (~176 A)= {MI, ..., M,}. Then H (~) = M1 " .-- " Mr H(~)(x)=MI(x)-....Mr(x), VxE X, and the latter product is direct. The following corollary justifies our claim that non-discrete, locally primitive groups behave like semisimple groups. Corollary 1.7.2. -- Let T = (X, Y) be a locally finite tree and H < Aut T a closed, non- discrete, locally primitive group. Then, H(~//QZ(H (~/) is a direct product of topologically simple groups. We turn now to a more precise geometric version of Corollary 1.7.2: let g := QZ(H(~/)\T be the quotient graph, q0 : T ---+ 0 the covering map and o) : Aut~T --~ Auto the associated homomorphism. We have the inclusion H C Autq, T and the group G:= 0~(H) __ H/QZ(H (~)) is closed and non-discrete in Auto; moreover, for every x E X, the homomorphism c0 induces an isomorphism H(x) -~ G(q~(x)) via which H_(x) < Sym E(x) is permutation isomorphic to _G(qo(x)) < Sym E(qa(x)). In particular, G is locally primitive as well, G (~) -~ H(~)/QZ(H (~/) and QZ(G (~)) = (e). Let {M,, ..., Mr} be the set of minimal closed normal subgroups of G (~/ (see Corollary 1.5.2); we know (Corollary 1.7.2) that G (~) = M1....-M, this product being direct. 130 MARC BURGER, SHAHAR MOZES Let m := M2 9 ... 9 Mr\l~ M:=image of M1 in Autm, q := M\m, p : m ---+ q the canonical projection x : NAutm(M) > Aut q, the canonical homomorphism. Corollary 1.7.3. 1) M < Aut m is closed, topologically simple, non-abelian. 2) NAutm(M) < Autm is locally primitive, and NAutm(M) (~ = M. 3) There is an isomorphism t~ ~ m Xp ... Xp m, of g with the r-fold fiber product of m with respect to p. 4) Identifying t~ with 1I~ I m under this isomorphism, we have: a) G(~) = M r b) M r < G < NAut~(M ~) C) NAutt~(M r) = (Nautm(M) x~ ... x~ Nautm(M)) N Sr. Applying Corollary 1.7.3 to locally 2-transitive groups, we obtain Corollary 1.7.4. -- Let H < AutT be a closed, non-discrete, locally 2-transitive subgroup. Then H(~ (~/) is topologically simple. This corollary is an analogue of the celebrated Theorem of Burnside, saying that the socle (that is the subgroup generated by all minimal normal subgroups) of a doubly transitive finite permutation group is either non abelian simple or elementary abelian (see [Di-Mo], Thm. 4.1B); observe moreover that in our case H(~//QZ(H (~/) is never abelian. 1.7.2. Here we collect two facts from Bass-Serre theory, and a corollary concerning locally quasiprimitive groups, which are of independent interest. For proofs of Lemmas 1.7.5 and 1.7.6 see the discussion in [Se] 4.4. Lemma 1.7.5. -- Let T = (X, Y) be a tree, a, b E X adjacent vertices, Ka, Kb subgroups of AutT fixing a, b, respectively, and acting transitively on E(a), E(b), respectively. Assume that Ka(b) = Kb(a) = K~ N Kb, let G:= (Ka, Kb>. Then G(a) = K~, G(b) = K b. GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 131 Lemma 1.7.6. -- Let T=(X, Y) be a tree, x C X, { yl,...,y~} the set of vertices adjacent to x, Kyl, ..., Kyu subgroups 0fAut T such that Kyi fixes yi and acts transitively on E(yi), 1 ~< i ~< d. Let Kx < Aut T, fixing pointwise E(x). Assume that K,~(x)= Kx, 1 <. i <<. d; let G:= (Ky~,..., Kyd). Then, G(yi) = K~i, 1 <<. i <. d, and G(x) = K~. Corollary 1.7.7. -- Let T = (X, Y) be a locally finite tree and H < AutT a closed, non-discrete locally quasiprimitive subgroup. Assume that for some x E X, H_(x) acts regularly on E(x). Then T(x, 1) is a precise fundamental domain for H (~/, in particular H(~/\T is a star. Proof. -- Let { Yl,...,Yd} be the set of vertices adjacent to x, Kyi:=H(yi), 1 ~< i~< d, K,:=HI(x) andy E {Yl,...,Yd}- Then H{x,y} =X. We claim that H_(y) does not act regularly on E(y); indeed, otherwise H(z) would act regularly on E(Z) for all z C X which would imply that Hi(x)= {e} and H is discrete, a contradiction. Thus X--Hx U Hy is the partition of X into H-orbits. For the subgroup G = (Kyl, ..., K,e), we obtain using 1.3.1. that X= G{y~, ...,yd, x}. Since Gx C Hx and G{yl, ...,Yd} C Hy we conclude that G{ yt,-..,yd } = Hy. This implies G = (K z : z E Hy) and shows that G is normal in H. On the other hand, G is nondiscrete, thus G D H (~/. Applying Lemma 1.7.6 to G, we obtain in particular that G(x) and hence H(~/(x) acts trivially on E(x); this proves the corollary using Lemma 1.4.2 and Proposition 1.2.1). [] 1.7.3 Proof of Theorem 1.7.1. -- For z C X, let Kz:=MI(z)..... My(z); observe that Mi N Mj C QZ(H(~/), Vi:~j, thus Mi(z) M Mj(z)=(e), Vi~:j, and hence [Mi(z), Mj(z) ] = (e), V i ~:j. For adjacent vertices x, y, define: Nx,y :-- NMI(x)(M1 (x,y) ) . ... . NMr(x)(Mr(x,y ) ). One verifies easily that Nx,y is a subgroup of H(x), normalized by H(x,y). Given a vertex x E X, we claim that either (1) N~,y C H(x,y) and K~(y)=Ml(x,y)-....Mr(x,y) for all verticesy adjacent to x, or (2) N~.y.H(x,y)--U(x) for all y adjacent to x, K~ acts transitively on E(x) and acts regularly on E(x), V 1 <~ i <~ r. Indeed, since H(x,y) is maximal in H(x) (see 0.1), we have either Nx,y.H(x,y) = H(x,y), or Nx,y.H(x,y)= H(x), for one and hence, by transitivity of H(x) on E(x), ally adjacent to x. In the first case, Nx,y C H(x,y) implies NMi(x)(Mi(x,y))=Mi(x,y), and Nx,y=Ml(x,y)..... Mr(x,y). Since on the other hand, K~(y)=Kx M H(x,y) C N~,y, we obtain Kx(y)=Ml(x,y)..... Mr(x,y), for ally adjacent to x. In the second case, Nx,y and hence Kx acts transitively on E(x). Moreover, Nx.y normalizes Mi(x,y) and hence _l_M~(x) 132 MARC BURGER, SHAHAR MOZES leaves invariant the fixed point set of Mi(x,y) in E(x), thus Mi(x,y) acts trivially on E(x) and since this holds for ally adjacent to x, we deduce that _~(x) acts regularly on E(x). Assume that (2) holds for some x E X. If (2) holds for some y adjacent to x, then, since all vertices are H-conjugate to x ory, we get that _Mi(z) acts regularly on E(Z) for all z E X and therefore Mi is discrete, a contradiction. Thus, if {yl, ...,y~} denotes the set of vertices adjacent to x, (1) holds foryl, ...,y~. In particular, we have Kyi(x)= Ml(Yi, X) . ... . Mr(Yi, X) , 1 <~ i <~ d. Since Me(x) acts regularly onE(x), 1 ~< g ~< r, we get Me(yi, x)=Me(yj, x), Vi,j and hence Kyi(x)=Kyj(X), Vi,j. Set B=Kyi(X) and define G:= (Kyl,...,Kyd). If gy i acts trivially on E(yi), then _Me (Yi) < Sym E(yi) is trivial and _M~ (x) < Sym E(x) is regular. Since any vertex is H-conjugate to x or yi, we deduce that Me is discrete, a contradiction. Thus Ky i does not act trivially on E(y/) and, being normal in H(yi), acts transitively on E(yi). We are thus in position to apply Lemma 1.7.6 and obtain G(yi)= Kyi, 1 ~< i ~< d, and G(x)=B; in particular G < AutT is locally closed and hence closed. Since T(x, 1) is a precise fundamental domain of G and +H preserves the bipartite structure of T, we have G{ys,...,y~} =+H{ys,...,yu } (see the proof of Cor. 1.7.7); thus, we have the equality G = (K z : z E +g{yl,---,Yd }), which implies that G is closed, normal in +H and hence G = H (~), since G acts non-freely on X. In particular B = G(x)= H(~/(x) D Kx which is a contradiction since B acts trivially on E(x) and Kx acts transitively on E(x). We conclude that N,,y C H(x,y) and K~(y) = Ms(x,y)..... Mr(x,y) for all pairs x,y of adjacent vertices. There are two cases: (1) At some vertex x E X, K~ induces the identity on E(x). By the same ar- gument as above, we conclude that Ky~ acts transitively on E(y/) and since Ky~(x)=K~(yi)=Kx(yj)=Kyj(X), we obtain, applying Lemma 1.7.6 to G = (Ky~, ..., Kyd) = H (~/, that G(z) = K z = HIll(Z), V z E X. (2) K~ acts transitively on E(x), Vx E X. Then, for G= (Kx, Ky), we have G(z)=K z by Lemma 1.7.5, thus G=H/~/ and again H(~/(z)=Kz, Vz E X. Finally we obtain that H(~)(x)= MI(x).....Mr(x) , V x E X; this implies that Ms....-Mr is open, hence closed in H(~/; on the other hand as this subgroup Ms "..." MT is normal in H, closed and non-discrete it must contain H/~/, we conclude that Ms "...'Mr = H (~/. [] Proof of CoroUary 1.7.2. --This is a direct consequence of Theorem 1.7.1 and Proposition 1.5.1 applied to A = QZ(H(~)). [] GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 133 Proof of Corollary 1.7.3. -- The fact that M is topologically simple, non abelian, fol- lows from Prop. 1.5.1 2a, 2c; this shows 1). We apply Theorem 1.7.1 with A= QZ(H(~)); let J/~,f (H (~/, A)= {N1,...,Nr} and Mi=N//A. Then dg,y (G (~ e)= {M1, ..., Mr}. Theorem 1.7.1 implies then that the hypotheses of Prop. 1.6.3 are satisfied. Using moreover that all M~, ...,Mr are G-conjugate we obtain by Prop. 1.6.3 an isomorphism ~ In Xp ... Xp m equivariant w.r.t, the isomorphism G (~ ---+ M r. Identifying g with 1-I~r)m and G (~ with M r, we obtain M r < G < NAutg(Mr), which shows 4a) and 4b. To show 4c) we observe that, since M is topologically simple and non-abelian, any continuous automorphism of M r permutes the factors. Given g E NAuto(M r) we may thus find 6 E Sr < NAuto(1V[ r) such that h = og fixes every factor of M r. Let q = G(~ let M~ be the product of all the factors except the /th one, let Pi : g -+ m = Mi\g and q : m --. q be the natural projection maps and n : Nautm(M) ---+ Autq the induced homomorphism. The element h = 6g induces via Pi an element hi E NAutm(M) and we have obviously x(hi)=l~(hj), Vi,j. The injection of r-r(r)~,T ~A~ 11~ xv~-] -"+ NAutg(Mr) sends then (hl,...,hr) to h=cig. This shows 4c). Concerning the assertion 2) we observe that G and hence NAut~(M r) = (r) (YL NAutm(M)) n Sr is locally primitive which implies, using Lemma 1.6.3 and the discussion preceding it, that NAutm(M) is locally primitive. [] Proof of Corollary 1.7.4. -- Using the notations of Corollary 1.7.3, assume that r >/ 2, and let x= (v, ..., v) E Vertg, v E Vertm, such that G(~ Then, we have the inclusion _G(x) < F r >4 Sr, where F=N(M)(v); this implies that the wreath product F r ~ Sr < Sym (E(v) r) is 2-transitive, this is impossible when r /> 2. Indeed given distinct points a, b in E(v) it is clear that no element of F r n Sr sends the pair ((a, a, a, ..., a), (a, b, b, ..., b)) to the pair ((b, b, b, ..., b), (a, b, b, ..., b)). Thus r= 1, and we conclude using Proposition 1.5.1 c). [] 1.8. Examples, the graph of diagonals Let T=(X, 3 0 be a tree, q the tree consisting of an edge and q0 : T ---+ q a morphism; the fiber product !~ := T x~ T is our graph of diagonals of T. Observe that Aut T permutes the fibers of ~0 and, let n : Aut T ~ Aut q denote the associated canonical homomorphism. Assume now that T is locally finite, let H < AutT be a closed, non-discrete, locally primitive subgroup and consider L := (H x~ H) x (z) where "~ E Aut (T x~0 T) is the automorphism given by the switch of factors. Let ~= 5~ be the universal covering tree and 1 > xl(--~) , G co >L , 1, 134 MARC BURGER, SHAHAR MOZES be the associated exact sequence. If H(x) < SymE(x) is non regular V x C X, we know (see Proposition 1.6.1) that L, and hence G is locally primitive; moreover L (~) = H (~) x H (~), and ~(_~) C QZ(G). Proposition 1.8.1. -- Assume that H < Aut T is closed, non-discrete, vertex transitive and that H (~/ is locally 2-transitive. Then (1) G < Auto~ is closed, non-discrete, locally primitive, and vertex transitive. (2) [G (~162 gl(-~)] =~1(-~). Remark 1.8.1. --It follows from (2) that G (~) D /~1(~) and hence r (~)) = G (~). The following is an interesting special case of the above situation: let p be an odd prime, ~+1 the Bruhat-Tits tree associated to PGL(2, O~) and detg 2 } H,:= g P L(2, Q,/:-- (Z;) ]detg] Then Hp < AutO+ l is vertex transitive, H~)= H; = PSL(2, Qp) is locally 2-transitive and we may apply Proposition 1.8.1. Notice that the image of a matrix g E GL(2, Qp) in PGL(2, Qp) is actually in PSL(2, Qp) exactly when the determinant of g is a square in Qp, hence H; is exactly PSL(2, Qp) and since PSL(2, Qp) is simple it follows that this is exactly H(p ~). In particular, we obtain an extension, 1 > ~1(5~p) > Op > (Hp � Hp) x (~) > 1 where ~p=~§ x~ ffp+l is the graph of diagonals of 37"p+1; this graph is regular of degree (p + 1) 2 and Gp < Aut(@§ It follows from Proposition 1.8.1 that nl(~p) C G~ ~162 hence G~)=G; since H~~ we obtain therefore an extension of PSL(2, Qp)2 1 , nl(!~p) , G; , PSL(2, Q~)x PSL(2, Qp) , 1 by the discrete group gl(!2~p), with the property In particular, since PSL(2, Qp)2 is perfect, we conclude that G; is perfect. Finall~ it is easy to see that ez( ) = ez(G;) = GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 135 To show this last assertion, observe that any g E QZ(Gp) is, modulo ~l(~p), of the form (hi, h2) with hi C QZ(Hp)= (e), which shows QZ(G~) C nl(~p). We conclude by observing that ~l(~p) C QZ(G;) C Qz(%). Proof of Proposition 1.8.1. --Let v=(x,x) be a vertex of~ :=T x~oT; the group G acts by conjugation on rc~(~, v) and, since T is a tree, the group rtl(~, v) is generated by all o-l(Hx~H)-conjugates of elements of n~(!~, v) represented by cycles of length four based at v. Such a cycle is given by a sequence of four consecutive edges (el, 4), (e2, e;), (e3, 4), (e4, d4), where either -- E (i) el=e2, e3=i4, e'l =e'4, e;=e'3 or D m (ii) el =e4, e2=g3, e'l=4, e~=e'4. Let ~ be a cycle of type (i) and e E Y with o(e)=x, e f[ {el, e3}; let I~ be the cycle given by (e, e'l) , (g, e;), (e3, ~), (g3, e'l--) and choose h = (hi, id) E H(~)(x) x H(~)(x) with hie3 =el, hlel =e3, hle=e. Observe that, since o~(G(~/) is normal and cocompact in L, it contains H (~/ x H (~). Thus, we may choose g E G (~ such that o~(g) and h coincide on a ball of radius 2 centered at v. A computation gives then gl3-1g-t[~ = 0~, thus [G (~), rc~(!~)] contains all paths of type (i) and, by a similar argument, an paths of type (ii). Since [G (~ rtl(~)] is normal in G D o-~(H x~ H), we obtain [G (~/, nl(~)] = ~1 (~)- [] 2. Thompson-Wielandt, revisited 2.1. Let T be a uniform tree, that is a locally finite tree such that Aut T\T is finite and AutT is unimodular. Then, T admits uniform lattices ([Ba-Ku]) and, guided by the analogy between trees and (rank one) symmetric spaces, one is lead to the question whether, given a Haar measure on AutT, there exists a constant c > 0 such that Vol (A\Aut T)/> c for all uniform lattices A C Aut T. As observed by Bass-Kulkarni ([Ba-Ku]), the answer for regular trees ~ turns out to be negative, provided n/> 3. In fact, if n is not a prime, one can even find an 136 MARC BURGER, SHAHAR MOZES increasing sequence Ai C Ai+l of (cocompact) lattices in Auto~', such that Ai\.~is an edge, and Vol(A/\Aut~) ---4 0, for i --+ +oc. There is, however, Conjecture (Goldschmidt-Sims). -- Given a locally finite tree T, there are only finitely many Aut T-conjugacy classes of locally primitive discrete subgroups. Equivalently, there should exist c > 0, such that Vol(A\AutT) >/ c for every locally-primitive lattice A C Aut T. This conjecture was settled for the 3-regular tree by Goldschmidt ([Go]). A result supporting this conjecture is the theorem of Thompson-Wielandt which says that there exists a neighbourhood U of the identity in AutT such that the intersection A N U is a nilpotent group for any locally primitive lattice A C AutT; notice the analogy with the Margulis Lemma (see [Bal-G-S]). More precisely, Theorem 2.1.1 (Thompson [Th], Wielandt [Wi]). -- Let T = (X, Y) be a locally finite tree and A < Aut T a discrete, locally primitive subgroup. Then a) A2(Z) is a p-group for some vertex Z E X and some prime p. b) /fA is vertex transitive and x,y are adjacent vertices, then Al(x,y) is a p-group for some prime p. If moreover Al(X,y) 89 then Op(Al(X)) ~ Al(x,y). This theorem is used in an essential way in Section 3, to ensure non-discreteness of certain locally primitive groups whose action on a sphere of radius 2 are known. For a proof of Thm. 2.1.1, the reader may consult [B-C-N] 7.2. In this section we will deduce Theorem 2.1.1 from a statement which is valid for all closed locally primitive groups. We introduce now a few notions and notations concerning profinite groups (see Lemma 2.2.0). For a profinite group K and a prime p, Op(K) denotes the unique maximal closed normal pro-p-subgroup of K, and OP(K) is the intersection of all closed normal subgroups N <] K, such that K/N is a pro-p-group. Thus K/OP(K) is the largest pro-p-quotient of K. Similarly, O~(K) denotes the unique maximal closed normal pro-solvable subgroup of K, while O~(K) is the intersection of all closed normal subgroups N <] K such that K/N is pro-solvable. Thus K/O~(K) is the maximal prosolvable quotient of K. A finite subgroup F < K is a component if it is quasi simple and subnormal (see [B-C-N] 7.1); finally E(K) is the closed subgroup generated by all components of K. Proposition 2.1.2. -- Let T=(X,Y) be a locally finite tree and H < AutT a closed, locally primitive subgroup. (1) a) ffH2(x) ~=(e), for some x C X, then there exists y C X such that E(H(y)) = (e). b) /fH acts transitively on X and Hl(x,y) ~ (e) for some pa# of adjacent vertices, then E(H(z) ) = (e), V z E X. GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 137 (2) a) /f O/H(x)) 7 ~ (e) for some x E X, then there exists y E X such that H2(y) is a pro-p group. b) If O~(H(x)) ~ (e) for some x E X, then there exists y C X such that H2(y) is pro-solvable. (3) Assume that H is vertex transitive and let x,y E X be adjacent vertices. a) If Op(H(x)) 7 ~ (e), then Hl(x,y) is pro-p. If moreover H~(x,y) 7 ~ (e), then Op(H,(x)) r Hl(x,y). b) If O~(H(x)) 5 ~ e, then Hl(x,y) is pro-solvable. If moreover H,(x,y) ~ (e), then O~(Hl(x)) r nl(x,y). Remark 2.1.1 (Prop. 2.1.2 implies Thin. 2.1.1). -- If F < AutT is discrete locally primitive, F(x) is a finite group for every x E X. Assume that F is vertex transitive; we show how to deduce Thm. 2.1.1 b). If Fl(X,y):~(e), then (1) b) E(F(z))--(e) V Z E X; on the other hand, F(x) is non-trivial, hence its generalized Fitting subgroup E(F(x) )- H Op(F(x)) is non-trivial as well, implying the existence of a prime p with Op(F(x))=~(e). Thm. 2.1.1 b) follows then from Prop. 2.1.2.3 a). Remark 2.1.2. -- If K is a profinite, non-finite group it can happen that E(K)-I-IpOp(K) = (e), and indeed there are (non-discrete) locally primitive groups H < AutT such that H2(x) is not pro-p for any p (see Section 3). 2.2. For ease of reference, we collect a few preliminary lemmas concerning profinite groups. Lemma 2.2.0. -- Let K be a profinite group. a) K admits a unique maximal closed normal pro-p-subgroup Op(K). b) If OP(K) denotes the intersection of all closed normal subgroups N of K with pro-p-quotient K/N, then K/OP(K) is the largest pro-p-quotient of K. Proof a) For any normal pro-p-subgroup N <~ K and any open normal subgroup D ~ K the image of N in the finite group K/D is a normal p-subgroup and hence is contained in Op(K/D), where for any finite group H, one denotes by O/H) the maximal normal p-subgroup of H (see [B-C-N] for a proof of the existence of Op(H)for finite groups). Define Op(K):=q~-l(IIo~K open Op(K/O)) where g) : K ~ I-[o ~ i~ open Op(K/O) is given by q)(k)= (kO)o < K. Since q0 is an isomorphism onto its image we deduce that Op(K) is a closed normal pro-p- subgroup. Moreover from the argument above it follows that it contains any closed normal pro-p-subgroup. 138 MARC BURGER, SHAHAR MOZES b) We have to show that K/(Y(K) is a pro-p-group. Let ~" denote the set of all closed normal subgroups N <1 K with K/N a pro-p-group. The homomorphism ~t : K ~ 1-INe~K/N, ~t(k) = (kN)Nej induces an isomorphism from K/OP(K) onto a closed subgroup of the pro-p-group I-[Nc~TK/N, which implies that K/OP(K) is a pro-p-group. [] Lemma 2.2.1. -- Let Hi < K, 1 <<. i <<. n, be closed subgroups with H1 <1 H2 <1 ... <1 Hn=K. a) If Hi/Hi_l is pro-p for all 2 <<. i <<. n, then OP(K) = OP(I-I ). b) IfHi/Hi_~ is pro-solvable for all 2 <~ i<<. n, then O~(K)=OC~(HI). Proof. -- Let us prove a), the proof of b) follows the same argument. Since Hi/Hi-1 is a pro-p group we have Hi_I D OP(Hi) and since Hi_I/OP(Hi) is a pro-p group we obtain that (Y(H/) D OP(H/_I). Since OP(Hi_I) is a topologically characteristic subgroup in Hi-i, we have O~ <1 Hi; as Hi/Hi_l and gi_l/OP(Hi_l) are pro-p groups we deduce that Ui/OP(Hi_l) is a pro-p group and hence that (Y(Hi_I) D (Y(Hi) and a) follows. [] Lemma 2.2.2. -- Let Hi < K, 1 <<. i <<. n, be closed subgroups with H1 <1 H2 <1 ... <~ Hn=K. a) Op(H,) C Op(K). b) Ooo(H1) C O~(K). Proof. --We shall establish a) (The proof of b) follows the same argument and is omitted). The subgroup Op(Hi) is a topologically characteristic subgroup of Hi, hence it is normal in Hi+~ which implies Op(Hi) C Op(Hi+l) and hence Op(Hl) C Op(K). [] For a profmite group K, the subgroups O~(K), CF(K), Ooo(K), O~(K), E(K) are all topologically characteristic, and we have OP(OP(K) ) = OP(K), O~176176 = O~(K). The following lemma is crucial and due to Wielandt in the context of finite groups. Lemma 2.2.3. -- Let Hi < K, 1 <~ i <~ n, be closed subgroups with H1 <1 H2 <1 ... <1 Hn=K. GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 139 Then a) Op(K) C NK((Y(H1)). b) Ooo(K) C NK(O~(H1)). c) E(K) C NK(H,). Proof. -- We begin with a general observation: let X = A.B be a group where A, B are subgroups of X, A <3 X, and Y < X with B <~ Y; then the homomorphism YNA , Y/B y ~---~ yB is surjective: indeed, fory E Y,y=a.b with a EA, b G B, thus a=yb -1 E YNA. a) Define X = Op(K)-(Y(HI), and L~ := X V1 Hi, 1 ~< i ~< n. We have OP(Ht) <1 Ll <:3 L2 <1 ... <1 L. =X. Since L1 N Op(K) ~ L1/OP(HI) is surjective, LI/OP(HI) is pro-p, and thus, by Lemma 2.2.1a) OP(OP(H1))=OP(L1), but OP(O~(H1))=OP(H~), thus OP(Ht)=OP(L1) and X=OP(K).OP(L1). Now we prove by recurrence that OP(L1)=O~(L/+I). Thus, assume (Y(L1) ..... OP(L/), so X=O~(K).(Y(Li); since Li <3 Li+I, Op(Li) <3 Li+I; since Li+l n Op(K) , Li+l/OP(Li) is surjective, L~+t/OP(Li) is pro-p, and hence OP(O~ thus OP(Li) = (Y(Li+I). Thus we conclude that OP(H~) ..... (Y(X) and in particular (Y(H~) <3 X, which implies Op(K) C NK(OP(H1)). b) Same argument, for "p = oc". c) See 7.1.3 (iv) in [B-C-N]. [] 2.3. Proof of Proposition 2.1.2. -- We first prove 3 a). Let x,y E X be adjacent vertices and assume that OP(Hl(x,y))~ (e); we have to show that Op(H(x))=e. Claim. -- For every z E X adjacent to y, we have (1) Op(H(y)) U Op(H(x,y)) C H(z). Indeed, pick a z --y violating (1) and let g E Op(H(y)) U Op(H(x,y)), with gz 7 ~ z. We have H~(z,y) <~ H~(y) < H(y) (2) Hl(z,y) <3 Hi(y) <3 H(x,y) 140 MARC BURGER, SHAHAR MOZES Lemma 2.2.3 a) implies that Op(H(y)) and Op(H(x,y)) normalize OP(HI(Z,y)), in particular, OP(H1 (z,y)) = gOP(H1 (z,y))g-1 = OP(Hl (gZ,y)). But OP(HI(z,y)) < H(z,y), (Y(HI(gz,y)) < H(gz,y), which implies, OP(HI(z,y)) <1 (H(z,y),H(gz,y))=n(y). Using an element x E H, interchanging Z andy we get OP(HI(Z,y)) <~ H(Z) and therefore OP(HI(z,y)) <? {H(z), H(y)) = +H (see 1.3.0 for the last equality) which implies that (Y(Hl(Z,y))=(e). Indeed we have that +g=(U(z),H(y))= H(z) *H(z,y)H(y) is a faithful amalgam. This proves the claim. Taking now intersection over all vertices z adjacent to y in (1), we obtain Op(H(y)) O Op(H(x,y))C Hi(y), in particular, Op(H(y)) C H(x,y) op(n(x,y)) <1 H,(y) <1 H(y) and thus by Lemma 2.2.2 and (1), Op(H(y)) C Op(H(x,y)) C Op(H,(y)) C Op(H(y)), which implies O~(H(y))= Op((H(x,y)) and Op(H(x,y))= Op(H(x)). Thus Op(H(y)) = Op(H(x)) <1 (H(x), H(y)) = +H and therefore Op(H(x))= (e). This proves the first part of 3 a). For the second part, assume that Op(H,(y)) C H,(x,y). Since Hl(x,y) <~ Hi(y) and Hl(x,y) is a pro-p-group, we have Hl(X,y) = Op((H,(x,y)) C O~(H~(y)) and thus Hl(x,y)=Op(Hl(y)), hence Hl(x,y) <~ H(y). Using an element exchanging x, y we obtain Hl(x,y) <1 (H(x), H(y))=+H which implies Hl(x,y)=(e) and concludes the proof of 3 a). The proof of 2 a) is the same, modulo replacing Hi(y, Z) by H2(z) in (2). GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 141 The proof of 2 b) and 3 b) works for "p = + oo" as above. The proof of 1 b) is analogous to the proof of 3 a) and is obtained by replacing Op(Hl(x,y)) by H~(x,y), O~(H(y)) by E(H(y)) and Op(H(x,y)) by E(H(x,y)). [] 3. From 2-transitivity to e~-transitivity In this chapter we consider vertex transitive, in particular 2-transitive groups of automorphisms of a (regular) tree and show that under additional assumptions of local nature, they are co-transitive; this is carried out in 3.3. In 3.1 we obtain some elementary properties of locally oc-transitive groups; we mention in passing that locally c~-transitive groups are quite well understood from the point of view of their unitary representation theory (see [Ne], [F-Ne], [B-M-]I , IN-M]2 ). In 3.2 we associate to every permutation group F < Sd the unique (up to conjugation) maximal vertex transitive subgroup U(F) of Aut Td acting locally like F; if F is 2-transitive these groups provide examples of oc-transitive groups. 3.1. We establish now a few basic properties of locally o~-transitive groups. Lernma 3.1.1. -- Let T = (X, Y) be a locally finite tree. For a closed subgroup H < Aut T, the following are equivalent: (1) H /s locally oo-transitive. (2) H(x) /s transitive on T(ec), V x E X. (3) H is non-compact and transitive on T(oc). (4) H is 2-transitive on T(oc). Any of these properties imply, (5) H_(x) < Sym E(x) /s 2-transitive and H is non-discrete. Proof. -- The equivalence 1) e=~ 2) and the implication 4) :=~ 3) are clear. For the equivalence 2) ~=~ 3), see [Ne]. Assume that 1) holds; let r : N ~ X be a geodesic ray and F, < Sym(E(r(0))\e), e= (r(0), r(1)), be the transitive permutation group defined by the restriction of N H(r(k)) to E(r(0)). The decreasing sequence F, > F~+I of finite k=0 transitive permutation groups stabilizes and hence F := N F, is transitive on E(r(0))\e. k=l 0(3 A compactness argument implies that the restriction to E(~(0))\e of N H(v(k)) coincides k=0 with F. This implies that the stabilizer in H of the end ~ E T(o~) defined by r acts transitively on T(cx~)\{{}. This shows 1) ~ 4). Finally the implication 1) =~ 5)is clear. [] 142 MARC BURGER, SHAHAR MOZES In view of the preceding lemma, locally a-transitive groups are locally primitive and we may apply Proposition 1.2.1. This leads to Proposition 3.1.2. -- Let T be a locally finite tree and H < AutT a closed, locally oo-transitive group. Then, (1) QZ(H) = (e). (2) H (~) /s locally e~-transitive and topologically simple. Proof. -- 1) Let S C T(oo) be the set of fixed points of hyperbolic elements in QZ(H); then S is countable and H-invariant, hence S = (~. Every g E QZ(H) 71H + , g :~ e, being hyperbolic, we deduce QZ(H)M H + =(e) and hence Iez(g)l .< 2, which finally implies QZ(H) C Z(H)= (e). 2) Since H is 2-transitive on T(oc) and (e) :~ H (~176 is normal in H, we conclude that H (~176 is transitive on T(e~); being also closed and non-compact, Lemma 3.1.1 implies that H (~176 is locally oc-transitive. Thus, by 1), QZ(H (~176 =(e); applying prop. 1.2.1, we have that (H(~/) (~) is a cocompact characteristic subgroup of H (~/. In particular it is cocompact and normal in H. Hence (H(~176 (~) = H (~176 Applying again Prop. 1.2.1 we deduce that H (~) is topologically simple. [] 3.2. -- The universal group U(F) Let d ) 3 and 4= (X, Y) be the d-regular tree. A legal coloring is a map i:Y ) {1,2,...,d}, such that: (1) i(y)= i(y), Vy E Y. (2) ilE(x/: E(x) ~ {1,2, ...,d}, is a bijection Vx E X. Given a permutation group F < Sd and a legal coloring i, the group U(,I(F ) = {g E aut~'ilE(g,)g i}E-(~ / E F, Vx E X} is a closed subgroup of Aut~. It enjoys the following basic properties: (1) U(,)(F)(x) < SyrnE(x) is permutation isomorphic to F < Sd, Vx E X. (2) The group U(,}(F) acts transitively on X. (3) Given legal colorings i, {, the corresponding subgroups U(,)(F) and U(/,)(F) are conjugate in Aut~. These properties follow from the fact (see [L-M-Z]) that a quadruple (il,/2, x~, x2) consisting of legal colorings i, { and vertices Xl, x~ E X determines a (unique) automorphism g E Aut~ with g(xl)=x2 and/2 = il og. GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 143 Henceforth we shall write U(F) without explicit reference to a legal coloring. The definition of U(F) implies readily that this group enjoys Tits' independence property (see [Ti]2 4.2); using [Ti]2 4.2 this observation implies then: Proposition 3.2.1. -- I) The group U(F) + is trivial or simple. 2) The group U(F) + /s of finite index in U(F) if and on~ if F < Sd is transitive and generated by its point stabilizers; in this case, U(F) + = U(F) ~ (Aut o~) + and is of index 2 in U(F). An important property is that, among all vertex transitive groups which are locally permutation isomorphic to F < Sd, the group U(F) is maximal; more precisely, Proposition 3.2.2. -- Let F < Sd be a transitive permutation group and H < Aut~ a vertex transitive subgroup such that H_(x) < Sym E(x) is permutation isomorphic to F < Sd. Then, for some suitable legal coloring, we have H < U(F). Proof. -- We construct an appropriate legal coloring: fax b E X and, for every x E X\{b}, let ex E Y be the unique edge defined by o(e,)=x, d(x, b)=d(t(ex), b) + 1; for every y E Y, choose Cy E H with CSy(y)=y. We define a legal coloring i: Y ~ {1,2, ..., d} inductively on En := t_J E(x) as follows: for n = 1, E1 = E(b), xCS(b, n-l) and we choose a bijection ib : E(b) ~ {1, 2, ...,d} such that ibH(b)ib 1 =F. Assume that i is defined in E,; for every x E S(b, n), define ilE(~):= ilE(y>~SexlE(~, where y = t(ex). One verifies easily that i:Y ~ { 1, ..., d} is a legal coloring and that H < U(,?(F). [] Now we turn to the structure of U(F)(x), x E X, in the case where F < Sd is transitive. Let A:={1, 2,...,d}, D := {2, ..., d} , An:=A x D n-1 , n ) 1, and F1 := StabF(1 ). Using the legal coloring i, one obtains a family of bijections bn : S(x, n) ,An, n >~ 1, uniquely determined by the following two properties: a) b~ : S(x, 1) ~ A, b~(y)=i((x,y)). b) rcnbn=bn-lp, where pn : S(x, n) ~ S(x, n - 1), nn : A,=An-I X D ~ A,_I denote the canonical projection maps. Define inductively F(n)<SymA, as follows: F(1)=F <SymA, and for n /> 2, F(n) = F(n - 1) ~ F~I "-l is the wreath product for the action of F(n - 1) on An_ 1. The bijection b, induces then a surjective homomorphism U(F)(x) --~ F(n) with kernel g(F)dx) := {g ~ g(F)(x) : gls(x, n) = id}. 144 MARC BURGER, SHAHAR MOZES The homeomorphism b~ : ff(e~) --+ lim An induces then an isomorphism U(F)(x)--~l'ma F(n) of topological groups. 3.3. In this section F < Se denotes a permutation group on d letters and H < Aut~d a closed vertex-transitive subgroup, such that H_(x) < SymE(x) is permu- tation isomorphic to F < Su. Let F 1 be the stabilizer in F of the letter 1. In this section, F < Sd will at least be 2-transitive. For a list of all 2-transitive (fmite) permutation groups, we refer to [Ca]; explicit realizations of most of them can be found in [Di-Mo], we have used these more detailed descriptions for obtaining the lists appearing in the examples in this section. In the following propositions and examples we use the notations of the Atlas [At] of finite groups. Proposition 3.3.1. -- Assume that F is 2-transitive and F1 is simple non-abelian. Then, H_l(x) ~-- F~ , where a E {0, 1, d}. Moreover, a E {0, 1 } e=~ H is discrete. a = d -: :- H = U(F). Remark. -- Let us note the following useful fact which we shall establish in the proof of Prop. 3.3.1: if x and y are adjacent vertices, Hl(x,y)/H2(x) ~ Fb~ with bE{0, d-1} and b--0 e=~ H is discrete. b = d- 1 .: :. H=U(F). Example 3.3.1. -- a) F < Sd, 2-transitive with non-abelian socle, and Fb simple non- abelian. d F F~ (n) 6) n An An-1 1 1 L2(1 1) A5 12 Mll L2(1 1) 12 M12 Mll 15 A7 L2 (7) 22 M22 L3(4) 23 M23 M22 24 M24 M23 GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 145 b) F < Su, 2-transitive of affne type, and F1 simple non-abelian. d F1 2 4 A6 2 4 A7 2 6 PSU(3, 3) 26m G2(2 m) Proposition 3.3.2. -- Assume that F < Sd is 2-transitive and Fl =S : 2 or S.2, where S is transitive, simple, non-abelian. Then, E(Hl(x)) --- S a where a C {0, 1, d}. Moreover, a E {0, 1} ~ H is discrete. a = d .'. :. H is oe-transitive. Example 3.3.2. d F F1 n S, Sn-1 (n >/6) 22 M22"2 La(4) : 29 176 HS U3(5) : 2 276 Co3 McL : 2 Proposition 3.3.3. -- Assume that F < Su is 2-transitive and F1 is quasisimple. Then, H,(x)/Z(H_,(x)) ~-- (F1/Z(FI)) a, where a C {0, 1, d}. Moreover, a E {0, 1} .'. > Hl(X,y)/H2(x) is abelian. a= d ,', :, H=U(F). Example 3.3.3. -- F < Sd is of affne ~pe. d FL q" SL(n, q) (n, q) + (2, 2), (2, 3) q2n Sp(n, q) (n, q) + (2, 2), (2, 3), (4, 2) 3 6 SL(2, 13) 3.4. In this section we collect a few simple facts used in the proof of the results of Section 3.3. 146 MARC BURGER, SHAHAR MOZES 3.4.1. Let T = (X, Y) be a locally finite tree, H < AutT a closed, vertex-transitive, locally primitive subgroup and x,y, z E X with x :~ z, adjacent toy. If Hl(x,y) C HI(z), then Hl(x,y)=(e) and H is discrete. Since +H has index 2 in H and +H\T is an edge, the locally compact group H is unimodular ([BK]). From Hl(x,y) C HI(Z) we deduce Hl(X,y) C Hi(y, z); these groups are conjugate in H, thus have the same Haar measure, which implies Hl(x,y)=Hl(y, z); this in turn implies Hl(X,y) <1 (H(x,y), H(y, z))=H(y), the latter equality following from the fact that H is locally primitive unless H_(y) acts regularly on E(y), in which case H_(t) acts regularly on E(t) for all t C X implying that H(t) is isomorphic to the finite group H_(t) and hence H is discrete. Using an inversion in H exchanging x,y we get Hl(X,y) <J (H(x), H(y)) = +H which implies Hl(X,y)= (e). 3.4.2. Let H < AutT be a 1-transitive (in particular vertex transitive) subgroup and n/> 1 with H.(x)=H.+l(X). Then H.(x)=(e). Indeed, for y E X adjacent to x we have Hn(x)=Hn+l(x) C H~(y), and applying an inversion which exchanges x,y we get H~(x)--H.(y). This in turn implies H.(x) <1 (H(x), H(y)); the latter group acts transitively on geometric edges, hence H,(x) = (e). 3.4.3. Let S <J L be finite groups, where L/S is solvable and S simple non- abelian. Let U < L n, n /> 1, with pri(U) D S for all 1 ~< i ~< n. Then pri(U Yl S")=S for all i, and U n S" is a (direct) product of subdiagonals of S n. Here, and in the sequel, a product of subdiagonals is a subgroup of S n of the form Ai~...Arr, where {Ij: 1 ~< j ~< r} is a partition of [1,n] and for any subset J c [1,n], Aj={(si) ESn : si=eVi~J, Sl=Sk Vl, kEJ}. Indeed, take k >/ 1 such that ~k(L)=S; we have then !~k(U) C S" and pri(5~k(U)) D S for all i, which implies that pri(U M S")=S for all i. The last assertion is then a well known fact concerning the cartesian powers of a non-abelian simple group. 3.5. In this section we state and prove the lemmas from which the results in 3.3 follow directl~ namely Prop. 3.3.3 follows from Lemmas 3.5.2 and 3.5.4, Prop. 3.3.2 follows from Lemma 3.5.1 and 3.5.3, Prop. 3.3.1 follows from Lemmas 3.5.1 and 3.5.3. Assumptions: in this section F < Sd denotes a transitive permutation group and H < AutTe a closed, vertex-transitive subgroup such that H_(x) < SymE(x) is permutation isomorphic to F. In the sequel, x,y E X always denote a pair of adjacent vertices. Lemma 3.5.1. -- Assume that F < Su is primitive and F1 = S, S : 2 or S.2, where S is simple non-abelian. Then E(H_I(x) ) -~ S a with a C {0, 1, d} and a E {0, 1 } if and only if H is discrete. GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 147 Proof. -- For y adjacent to x, the inclusion Hi(x) C H(x,y) gives rise to a homomorphism q0y : Hi(x) ---+ H(x,y)/Hl(y):=Hx,y --~ F1 with normal image, and an injective homomorphism q): _H,(x) ' Hyes(x, 1)Hx,y ~- ~" Since Imq0y <1 Hx,y and HI(x) <~ H(x), we have either %(Ul(x))=(e), Vy E S(x, 1) or q0y(Ul(x)) D E(Ux,y)=:Sy _~ S, Vy C S(x, 1). In the first case we obtain H_l(X)=(e), hence Hi(x)= (e) (by 3.4.2) and H is discrete. In the second case, it follows from (3.4.3) that q0(H~(x))N Hyes(x, 1)Sy is a product of subdiagonals; these subdiagonals determine a bloc decomposition for the H(x)-action on S(x, 1); since this action is primitive, there are two cases: in the first case we obtain the full diagonal; hence E(H_I(x)) -~ S and Hl(x,y)/H2(x) is a 2-group. In particular, if z C S(x, 1), z :~y, the image of Ht(x,y) in H(x, z)/HI(z) --~ FI is a subnormal 2-group, and hence trivial. Thus Hl(x,y) C H~(Z) which by 3.4.1 implies Hl(x,y)=(e) and H is discrete. Finally, in the second case we obtain E(HI(x)) D S d, in particular Hl(X,y)/H2(x) cannot be a p-group and hence H is not discrete by Thompson-Wielandt (Theorem 2.1.1). [] Lemma 3.5.2. -- Assume that F < Sd is primitive and Fl is quasisimple. Then H I(x)/Z(H_,(x)) ~- (F1/Z(F1)) a , with a E {0, 1, d}. Moreover, a E {0, 1} .: :. H,(x,y)/H2(x) is abelian. a = d -' :- Hk(x) /s non-abelian V k >1 1. Proof. -- The proof of Lemma 3.5.2 is completely analogous to that of Lemma 3.5.1; the only point which needs to be verified is that Hl(x,y)/H2(x) is abelian, if and only if _Hk(x) is abelian for some k/> 1. Assume Hl(x,y)/H2(x) is abelian; this holds then for an pairs x,y of adjacent vertices. For every z E S(x, 2) let p(z) C S(x, 1) be such that z is adjacent to p(z). Then, the injection H3(x) --+ HI(Z,p(z)), z E S(x, 2) gives rise to an injective homomorphism H~ 3 (x) ) H (HI(Z,P(z))/H2(z)) z~S(x, 2) which shows that H_~(x) is abelian. Conversely, assume that H_k(x) is abelian for some k/> 1. Then, for all m >1 k, H__m(X) injects into H _Hk(z), where the product is over all z E S(x, m - k). This implies that Hk(x) is prosolvable, thus O~(H(x)) ~: (e), and hence (Proposition 2.1.2 3) b)) Hl(x,y) is prosolvable, in particular Hl(x,y)/H2(x) is solvable, and since F1 is quasisimple, this implies that Hl(x,y)/H2(x) is abelian. [] In the next lemmas, c(n) denotes the cardinality of the sphere S(x, n) in o~a. 148 MARC BURGER, SHAHAR MOZES Lemma 3.5.3. -- Under the assumptions of Lemma 3.5.1 assume moreover that S is transitive on {2,..., d} and a = d. Then, E(Hk(x) ) "~ S ~(k) , V k >t 1. In particular H is oc-transitive. Proof. Notation. -- For z E S(x, n), set p(z) = S(x, n- 1)AS(z, 1), H~. z:= H(Z, P(z))/HI(z); for y E S(x, 1), set Sn(x,y):= {Z E S(x, n) : d(z, x)=d(z,y) + 1}, a(n)= ISn(x,y)l, c(n)= IS(x, n)]. First we claim that H is locally oo-transitive. Since H is non-discrete (Lemma 3.5.1), we have Hn(x) 4:(e) and hence (3.3.2) Hn(x) ~:Hn+~(x) for all n /> 1. Thus, for every n/> 1, there exists z E S(x, n) such that the image In. z of Hn(x) in Hx, z is non- trivial; let x0--x, Xl, ..., Xn-1 =p(z), Xn = Z, be the consecutive vertices of a geodesic path connecting x to Z, then: H,(x) <1 Hl(xl,...,x,_l) <1 Hl(xn-l) <1 H(p(Z), z), and therefore I,, z is a non-trivial, subnormal subgroup of Hi, z ~- F1, which implies I,, z D E(Hx, z) and hence is transitive on E(Z)\(z, p(z)). Using that F is 2-transitive and recurrence on n/> 2 one concludes that H(x) is transitive on S(x, n). Next we claim that if {Bi}~<,i,<k is a bloc decomposition for the H(x)-action on S(x, n) (n/> 2) with IBi N S,(x,y)l ~< 1 for all i andy E S(x, 1), then IBil = 1 for all i. Since H(x) acts transitively on S(x, n), all blocs have the same cardinality. Assume IBll /> 2 and choose y :~y' in S(x, 1) such that B1 N S,(x,y)= {Z}, B1 f-1 S,(x,y')= {z'}; choose t E S,(x,y') with t4:z', so that t E Bi for some i4: 1. Since H is locally oo-transitive we may choose g E H(x) with g(z) = z and g(z') = t;, this implies g(B1) = Bl and g(B1) = Bi, contradicting i 4: 1. This establishes the claim. Now we show by induction on n/> 1 that E(IzI,(x)) -~ S~('); this holds for n = 1. Let n >1 2; for y E S(x, 1), we have H,(x) <a H,_l(y), and, if In,x,y denotes the image of Hn_l(y) in Hz~s,(x,y)Hx, z, we have E(I,,x,y)~ S a(n) by the induction hypothesis; the same holds therefore for the image of H,(x) in l-Izes,(x,y)Hx, z. For the image H_,(x) of H,(x) in l-lyEs(x, 1) l--lzcs,i,,y)H,,z, we have that E(H_,(x)) is a product of subdiagonals in S4n); this induces then a bloc decomposition of S(x, n) as above, and hence E(_Hn(x)) ~-- S c("). [] Lemma 3.5.4. -- Under the assumptions of Lemma 3.5.2, assume moreover that F < Sd is 2-transitive and a = d. Then, H k(x) ~-- F~ k) , V k ~> 1. Proof. -- The proof proceeds as in Lemma 3.5.3, with the following additional remark: GROUPS ACTING ON TREES: FROM LOCAL TO GLOBAL STRUCTURE 149 By Lemma 3.5.2, there is for every n/> 1, a z E S(x,n) such that I,, z <~ Hx, z -~ F1, is non-abelian, and therefore I,, z--F1. From this one concludes that H is locally oo-transitive, and concludes the proof as in Lemma 3.5.3. [] REFERENCES [At] J. H. CONWAY, R. T. CURTIS, S. P. NORTON, R. A. PARKZR, R. A. WILSON, Atlas of finite groups, Oxford, Clarendon Press, 1985. [Bo-Ti] A. BOREL, J. TITS, Homomorphismes "abstraits" de groupes algdbriques simples, Ann. of Math. 97 (1973), 499-571. [B-C-N A. E. BROUWER, A. M. COHEN, A. NEUV~AIER, Distance Regular Graphs, Ergebnisse, 3. Folge, Band 18, Springer 1989. [B-~ M. BURaER, S. MozEs, Finitely presented simple groups and products of trees, C.R. Acad. Sci. Paris, t. 324, Serie I, 1997, p. 747-752. M. BUROER, S. MozEs, Lattices in products of trees, Publ. Math. IHES 92 (2001), 151-194. 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Publications mathématiques de l'IHÉS – Springer Journals
Published: Aug 30, 2007
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