Products of Trees, Lattices and Simple Groups

Shahar Mozes

1991 Mathematics Subject Classification: 20E06, 20E08, 20E32, 20F32, 22D40, 22E40

The group of automorphisms of a locally finite tree, denoted Aut T, is a locally compact group which exhibits behavior analogous to that of a rank one simple Lie group. This analogy has motivated many recent works, in particular the study of lattices in Aut T by Bass, Kulkarni, Lubotzky and others. Recall that 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. Motivated by this we study in several joint works with Marc Burger and with Robert J. Zimmer cocompact lattices in the group of automorphisms of a product of trees, or rather in groups of the form Aut T_{1} × Aut T_{2} where each of the trees T_{i}, i = 1, 2, is (bi-)regular. The results obtained concerning the structure of lattices in Aut T_{1} × Aut T_{2} enable us to construct the first examples of finitely presented torsion free simple groups, see [BM2]. One Tree. There is a close relation between certain simple Lie groups and groups of tree automorphisms. Let G be a simple algebraic group of rank one over a nonarchimedean local field K. Considering the action of G on its associated Bruhat-Tits tree T, we have a continuous embedding of G in Aut T with cocompact image. In [Tit], Tits has shown that if T is a locally finite tree and its automorphism group Aut T acts minimally (i.e. without an invariant proper subtree and not fixing an end) on it, then the subgroup Aut^{+}T generated by edge stabilizers is a simple group. In particular the automorphism group of a regular tree is virtually simple. These results motivated the study of Aut T taking this analogy with rank one Lie groups as a guideline, see [Lu1], [Lu3].

When T is a locally finite tree its automorphism group is locally compact. Recall that a subgroup Γ of a locally compact group G is called a lattice when it is discrete and the quotient Γ\G carries a finite invariant measure. In case the quotient is compact the lattice is called uniform. Observe that a subgroup of Aut T is discrete if and only if it acts with finite stabilizers. One may determine whether a discrete subgroup is a lattice by checking the finiteness of the sum ∑_{v∈F} 1/|Γ_{v}|, where Γ_{v} is the stabilizer of the vertex v and the set F is a fundamental domain for the action of Γ on some Aut T orbit in T. Of particular interest is the case when Aut T acts with finitely many orbits on T; in this case a lattice is uniform if and only if the quotient Γ\T is finite. Such lattices, called “uniform tree lattices”, correspond to finite graphs of groups in which all vertex and edge groups are finite. These were extensively studied by Bass and Kulkarni, [BK]. By a result of Leighton, cf. [BK], any two uniform tree lattices in Aut T are commensurable up to conjugation. A key role in the study of lattices in semisimple Lie groups is played by their commensurators (cf. [AB]): Comm_{G}(Γ) = {g ∈ G : g^{−1}Γg ∩ Γ is of finite index in both Γ and g^{−1}Γg}. In particular, Margulis has shown that an irreducible lattice Γ < G in a semisimple Lie group is arithmetic if and only if its commensurator is dense in G.

It was shown by Liu [Liu] (confirming a conjecture by Bass and Kulkarni) that the commensurator of a uniform tree lattice is dense. The situation concerning non-uniform lattices is much more involved and not well understood. There are examples, by Bass and Lubotzky [BL2], cf. also [BM1], of non-uniform lattices in the automorphism groups of regular trees whose commensurators are discrete. At the other extreme it was shown by the author that the commensurator of the Nagao lattice SL_{2}(F_{p}[t]) in the full automorphism group of the (p + 1)-regular tree is dense. An example of a cocompact lattice with dense commensurator which is not a uniform tree lattice appears in [BM1].

Note that as all uniform tree lattices of a given tree are commensurable up to conjugation, the isomorphism class of the commensurator of a uniform tree lattice is determined by the tree. In the other direction it is shown in [LMZ] that for regular trees the commensurator determines the tree. In proving this we use a superrigidity theorem for the commensurators of lattices in the automorphism groups of regular trees. In a much more general setting of divergence groups in the isometry group of CAT(-1) spaces we have shown in [BM1] (see also [Bur]) that:

Theorem 1. Let X, Y be proper CAT(-1)-spaces, Γ < Is(X) a discrete divergence group, Λ < Is(X) a subgroup such that Γ < Λ < Comm_{Is(X)}(Γ) and π : Λ → Is(Y) a homomorphism such that π(Λ) acts convex-minimally and π(Γ) is not elementary. Then π extends to a continuous homomorphism

πext : \overline{Λ} → Is(Y) .

Products of trees and locally primitive groups. Among the most striking results concerning lattices in semisimple Lie groups are the arithmeticity and superrigidity theorems established by G.A. Margulis (cf. [Mar], [Zim], [AB]). These assert that:

1. An irreducible lattice in a higher rank (i.e. ≥ 2) semisimple Lie group is arithmetic.

2. Any linear representation of such a lattice with unbounded image essentially extends to a continuous representation of the ambient Lie group.

Recall that a lattice Γ < G in a semisimple Lie group is called reducible if the following equivalent conditions hold:

1. There exists a decomposition of G (up to isogeny) as a product G = G_{1} ×G_{2} with both G_{i} non compact semisimple Lie groups and Γ projects discretely on each G_{i}, i = 1, 2.

2. Γ contains a finite index subgroup of the form Γ_{1} × Γ_{2} where Γ_{i} < G_{i} is a lattice and G = G_{1} × G_{2} a decomposition as above.

Using Borel’s density theorem ([Bor], cf. [Fur], [Dan]) we have that a lattice Γ in a semisimple Lie group G is irreducible if it satisfies the following equivalent conditions:

• (Ir1) The projection of Γ on any factor G_{i} of G with ker : G → G_{i} noncompact has non-discrete image.

• (Ir2) A projection as above has dense image.

Pursuing further the analogy between Aut T and rank one Lie groups, it is natural to ask for a structure theory for lattices in groups of the form Aut T_{1} × Aut T_{2} with T_{i} trees. In particular one would like to have “rigidity-” and “arithmeticity-” like results. Some steps in this direction were taken jointly with M. Burger and R.J. Zimmer [BMZ], [BM2], [BM3] Let us assume henceforth (unless explicitly stated otherwise) that our trees are (bi)-regular. A lattice Γ < Aut T_{1} × Aut T_{2} is reducible when its projections on each factor are discrete. Restricting our attention to uniform (i.e., cocompact) lattices observe that the projection pr_{i}(Γ) < Aut T_{i} of a lattice Γ < Aut T_{1} × Aut T_{2} is never dense. This follows by observing that the compact open subgroup K = Stab_{Aut T_{i}}(x), x ∈ T_{i} a vertex, maps onto (ℤ/2ℤ)^{ℕ} and hence is not topologically finitely generated, whereas the intersection of Γ with the product of K with Aut T_{3−i}, being a uniform lattice in this product, is finitely generated. Thus the intersection of K with the projection of Γ to Aut T_{i} cannot be dense in K (see [BM3]). Given a lattice Γ < Aut T_{1} × Aut T_{2} denote by H_{i} = \overline{pr_{i}(Γ)}, i = 1, 2. Thus Γ < H_{1} × H_{2}. Clearly the representation theory of Γ cannot be more rigid than that of H_{1} × H_{2}. Indeed, one can construct irreducible lattices Γ < Aut T_{1} × Aut T_{2} such that the corresponding groups H_{i} surject onto free groups. Requiring various conditions on the projections of Γ leads to interesting structure theory.

Definition 2. A subgroup H < Aut T is called locally primitive if for every vertex x ∈ T its stabilizer in H induces a primitive permutation group, denoted \underline{H}(x), on the set E(x) of neighbouring edges.

The class of closed locally primitive subgroups of Aut T has a structure theory reminiscent in some ways to that of semisimple Lie groups. A key role in the study of these groups is played by the following lemma which shows that normal subgroups of a locally primitive group are either free or very large.

Lemma 3. Let T be a tree and let H < Aut T be a closed locally primitive subgroup. Any normal subgroup N ◁ H either acts freely on T, or is cocompact and has a fundamental domain which is either a ball of radius 1 or an edge in T.

For a locally compact, totally disconnected group H, let H^{(∞)} := ∩_{L<H}L,

where the intersection is taken over all open subgroup L < H of finite index, and QZ(H) := ∪_{U<H}Z_{H}(U), where the union is taken over all open subgroups U < H.

Thus QZ(H) = {h ∈ H : Z_{H}(h) is open}. Observe that H^{(∞)} = ∩_{N◁H}N, where the intersection is taken over all closed, cocompact normal subgroups N ◁H. Note also that any discrete normal subgroup of H is contained in QZ(H). Using Lemma 3 one shows that when H is non-discrete, H^{(∞)} is cocompact in H.

With the (limited) analogy between closed locally primitive subgroups H < Aut T and algebraic groups G in mind, we may view H^{(∞)} < H as playing the role of the subgroup G^{+} < G generated by all one parameter unipotent subgroups. We have the following structure theorem:

Theorem 4. (Burger-Mozes) Let H < Aut T be a closed, non-discrete, locally primitive subgroup. Then H^{(∞)}/QZ(H^{(∞)}) decomposes as a finite direct product

H^{(∞)}/QZ(H^{(∞)}) = M_{1} · M_{2} · · · · · M_{r}

Where each M_{i}, 1 ≤ i ≤ r is a topologically simple group.

Various examples of closed subgroups of Aut T may be obtained via the following construction: Let d ≥ 3, and F < S_{d} be a permutation group. Let T_{d} = (X, Y) be the d-regular tree and i : Y → {1, 2, …, d} a legal (edge) coloring, that is, a map such that i(y) = i(ȳ), ∀y ∈ Y, and i|_{E(x)} : E(x) → {1, 2, …, d} is a bijection, ∀x ∈ X. Define U(F) = {g ∈ Aut T_{d} : i|_{E(gx)}gi^{−1}|_{E(x)} ∈ F, ∀x ∈ X}. Observe that U(F) is a closed subgroup of Aut T_{d}; the group U(F) acts transitively on X; the finite group \underline{U(F)}(x) < Sym E(x) is permutation isomorphic to F < S_{d}, and hence, when F is a primitive permutation group, U(F) is locally primitive. We notice also the following:

1. Using Tits’ theorem, [Tit], it follows that U(F)^{+} (the subgroup generated by edge stabilizers) is simple.

2. The subgroup U(F)^{+} is of finite index in U(F) if and only if F < S_{d} is transitive and F is generated by its subgroups Stab_{F} (j), 1 ≤ j ≤ d. In this case, U(F)^{+} = U(F) ∩ Aut^{+}T_{d} and is of index 2 in U(F).

3. Let F < S_{d} be a transitive subgroup and H < Aut T_{d} be a vertex-transitive subgroup such that, for some x ∈ X, \underline{H}(x) < Sym E(x) is permutation isomorphic to F < S_{d}. Then, for some suitable legal coloring, we have H < U(F).

We are especially interested at those subgroups U(F) which arise as closures of projections of irreducible uniform lattices. As these must be topologically finitely generated we note:

Proposition 5. [BM3] Let F < S_{d} be a transitive permutation group. Then U(F)(x) is topologically finitely generated if and only if F_{1} = Stab_{F} (1) is perfect and equal to its normalizer in F.

Notation: Denote by S(x, n) the sphere of radius n around a vertex x ∈ T. For H < Aut T, x ∈ X, n ≥ 1, H_{n}(x) = \{h ∈ H : h|_{S(x,n)} = id\}, \underline{H}_{n}(x) = H_{n}(x)/H_{n+1}(x).

Proposition 6. Let F < S_{d} be a 2-transitive permutation group such that F_{1} = Stab_{F} (1) is non-abelian simple and H < Aut T_{d} a closed vertex transitive subgroup such that \underline{H}(x) < Sym E(x) is permutation isomorphic to F < S_{d}. Then \underline{H}_{1}(x) ≃ F_{1}^{a} where a ∈ {0, 1, d}. Moreover

H is discrete ⇔ a ∈ {0, 1}
H = U(F)  ⇔ a = d.

In the proof of the above proposition one needs to show that when a = d the group H is not discrete. This is established using the Thompson-Wielandt theorem (see [Tho], [Wie], [Fan]).

Theorem 7. (Thompson-Wielandt) Let T = 𝔗_{n} be the n-regular tree. Let U < Aut (T) be the pointwise stabilizer of a ball of radius 1 around an edge e. (Note that U is an open compact neighborhood of the identity.) Then for every vertex transitive locally primitive lattice Γ < Aut (T) the group Γ ∩ U is an l-group for some prime l < n.

In the context of lattices Γ < Aut T_{1} × Aut T_{2} one would like to verify for a given lattice whether it is reducible or not, namely whether its projections are discrete or not. The Thompson-Wielandt theorem may be used to verify nondiscreteness and hence irreducibility in certain cases but we do not know a general algorithm for deciding this question.

Rigidity. The following result may be viewed as an analog of the Mostow rigidity theorem:

Theorem 8. [BMZ] Let Γ < Aut T_{1} × Aut T_{2} and Γ′ < Aut T′_{1} × Aut T′_{2} be uniform lattices. Assume that the subgroups H_{i} = \overline{pr_{i}(Γ)} < Aut T_{i}, i = 1, 2 are locally primitive. If we have an isomorphism Γ ≅ Γ′, then it is induced by an isometry between T_{1} × T_{2} and T′_{1} × T′_{2}.

Note that we do not assume that the lattices are irreducible. Bass and Lubotzky [BL1] have shown that a certain class of closed (non-discrete) subgroups of Aut T determines the tree T (up to some natural modifications). We note that any isomorphism between locally primitive lattices Λ < Aut T and Λ′ < Aut T′ acting without inversion on the corresponding trees is induced by an isometry between T and T′. (To establish this one notes that the tree structure may be reconstructed from, say, Λ using the correspondence between the vertices and maximal finite subgroups of Λ and between the edges and pairs of such maximal finite groups which generate the group and whose intersection is a maximal subgroup in each.)

Theorem 9. [BMZ] Let Γ < Aut T_{1} × Aut T_{2} be a uniform lattice with H_{i} = \overline{pr_{i}(Γ)} < Aut T_{i}, i = 1, 2 locally primitive. Let Y be a proper CAT(-1) space and π : Γ → Is(Y) a homomorphism such that π(Γ) is not elementary and acts convex-minimally on Y. Then π extends to a continuous homomorphism π_{ext} : H_{1} × H_{2} → Is(Y) which factors via a proper homomorphism of one of the H_{i}, i = 1, 2.

In the Lie groups setting, Margulis’ superrigidity theorem plays a key role in showing that irreducible lattices in higher rank groups are arithmetic. In the context of lattices in Aut T_{1} × Aut T_{2} we have:

Theorem 10. [BMZ] Let Γ < Aut T_{1} × Aut T_{2} be an irreducible cocompact lattice. Assume that each H_{i} = \overline{pr_{i}(Γ)} < Aut T_{i} is locally primitive. Then one of the following possibilities holds:

1. Every linear image of Γ is finite.

2. Γ has an infinite linear image over a field of characteristic 0. Then H_{i} is a p_{i}-adic analytic group for some prime p_{i}. The adjoint map, which we denote by φ = φ_{1} × φ_{2} : H_{1} × H_{2} → Aut (Lie(H_{1})) × Aut (Lie(H_{2})), yields a continuous surjection from H_{1} × H_{2} onto a semisimple Lie group over some local fields and the image φ(Γ) is an arithmetic lattice. Moreover, the kernel of this homomorphism is a torsion free discrete subgroup of H_{1} × H_{2}.

3. Γ has an infinite linear image over a field of positive characteristic p. Then there is a continuous map with unbounded image from H_{1} ×H_{2} into a simple Lie group over F_{p}((t)).

Let us remark that:

• It seems reasonable to expect in case 3 of the theorem a result similar to that of 2.

• In case 2:

– We do not claim that the image φ(Γ) is an irreducible arithmetic lattice. Indeed, one can construct examples where this lattice is reducible.

– The algebraic groups φ_{i}(H_{i}) need not be of rank one. In fact they are of rank one if and only if ker φ = {e}. Moreover, φ_{i}(H_{i}) is of rank one and ker φ_{i} is trivial exactly when the action of H_{i} on the corresponding tree T_{i} is locally infinitely transitive, i.e., the stabilizer of each vertex acts transitively on simple paths of arbitrary length starting at the vertex.

An example of an irreducible lattice in Aut T_{1} × Aut T_{2} which is an extension of an arithmetic lattice in a semisimple algebraic group G = G_{1} × G_{2}, where each G_{i} is a semisimple group over some local field k_{i}, may be obtained as follows, cf. [BM3]. Associated with each G_{i} one has an affine building Δ_{i} on which the group G_{i} acts. “Draw” on Δ_{i} a graph G_{i} defined in an equivariant way (for example let G_{i} = SL_{3}(ℚ_{p}), the associated Bruhat-Tits building is a simplicial complex whose set of vertices has a natural 3-coloring (see [Bro]); consider the graph consisting of the vertices belonging to two fixed colors and the corresponding edges). The group G acts on G_{1} × G_{2} and hence an extension H_{1} × H_{2} of G by π_{1}(G_{1} × G_{2}) acts on the universal covering space T_{1} × T_{2} of G_{1} × G_{2}. Taking a lattice Λ < G, its extension by π_{1}(G_{1} × G_{2}) is a lattice in H_{1} × H_{2}.

We examine next the normal subgroups structure of lattices in Aut T_{1} × Aut T_{2}.

Proposition 11. [BM3] Let T_{1}, T_{2} be locally finite trees, Γ < Aut T_{1} × Aut T_{2} a discrete subgroup such that Γ \ (T_{1} × T_{2}) is finite and N < Γ a normal subgroup such that the quotient graphs pr_{i}(N) \ T_{i}, i = 1, 2, are finite trees. Then Γ/N has property (T).

Proposition 12. [BM3] Let T_{1}, T_{2}, Γ be as in Proposition 11 and H_{i} := \overline{pr_{i}(Γ)} < Aut T_{i}.

(a) The homomorphism Hom_{c}(H_{1} × H_{2}, ℂ) → Hom(Γ, ℂ) mapping χ to χ |_{Γ} is an isomorphism.

(b) Let (π, V) be an irreducible finite dimensional unitary representation of Γ with H^{1}(Γ, π) ≠ 0.

Then π extends continuously to H_{1} × H_{2}, factoring via one of the projections.

The following result, obtained in [BM3], is an analog of Margulis’ normal subgroup theorem:

Theorem 13. Let Γ < Aut T_{1} × Aut T_{2} be a cocompact lattice such that H_{i} := \overline{pr_{i}(Γ)} is locally ∞-transitive and H_{i}^{(∞)} is of finite index in H_{i}, i = 1, 2. Then, any non-trivial normal subgroup of Γ has finite index.

The proof of this theorem follows the lines of the proof by Margulis of the corresponding result in the context of Lie groups. In particular one uses the following analog of the Howe-Moore theorem concerning vanishing of matrix coefficients. In the context of Aut^{+}T with T a regular tree this was shown in [LM], see also [FTN].

Theorem 14. [BM3] Let H < Aut T be a closed locally ∞-transitive subgroup and (π, H) be a continuous unitary representation of H with no nonzero H^{(∞)} invariant vectors. Then for every u, v ∈ H, lim⟨π(g)u, v⟩ → 0 as g ∈ H tends to ∞.

However there is an interesting application of Theorem 13 which is based on a fundamental difference between cocompact lattices in Aut T_{1} × Aut T_{2} and cocompact lattices in Lie groups. Whereas any finitely generated subgroup of a linear group is residually finite, finitely generated subgroups, and even cocompact lattices, in Aut T_{1} × Aut T_{2} need not be residually finite. A criterion for establishing that certain lattices are not residually finite is provided by the following:

Proposition 15. [BM3] Let G_{i}, i = 1, 2 be closed locally compact groups. Let Γ < G_{1}×G_{2} be a discrete subgroup. Assume that for i = 1, 2, G_{i}^{(∞)} < \overline{pr_{i}(Γ)} < G_{i}. Then

Γ^{(∞)} > [G_{1}^{(∞)}, Λ_{1}] · [G_{2}^{(∞)}, Λ_{2}],

where Λ_{1} := pr_{1}((G_{1} × e) ∩ Γ), Λ_{2} := pr_{2}((e × G_{2}) ∩ Γ). In particular, if each G_{i} has trivial centralizer and Λ_{1} × Λ_{2} ≠ e, then Γ is not residually finite.

In particular, let Γ < Aut T_{1} × Aut T_{2} be an irreducible lattice such that each H_{i} = \overline{pr_{i}(Γ)} is locally primitive. Each H_{i}^{(∞)} acts on T_{i} with finite quotient and hence has trivial centralizer. Thus if in addition the projection of Γ to one of the factors Aut T_{i} is not injective then Γ is not residually finite. The construction described following Theorem 10 provides a non residually finite lattice.

Combining Proposition 15 and Theorem 13 allows us to construct (see [BM2] and [BM3]) examples of finitely presented torsion free simple groups. One constructs first a non residually finite lattice Γ̃ which satisfies the conditions of Theorem 13. Given Γ̃ let Γ = Γ̃^{(∞)}. It follows that Γ is the minimal finite index subgroup of Γ̃. Verifying that Γ satisfies the conditions of Theorem 13, one deduces that Γ is simple. It should be observed that a lattice Γ which satisfies the conditions of Proposition 15 for not being residually finite must have a non trivial normal subgroup of infinite index! namely either Λ_{1} or Λ_{2}. Thus in order to produce a non residually finite lattice which satisfies the conditions of Theorem 13 one uses the geometric description of lattices in Aut T_{1} × Aut T_{2} (see below) to embed a non residually finite lattice obtained using Proposition 15 in a lattice as in Theorem 13.

Theorem 16. For every pair (n, m) of sufficiently large even integers there exists a group Γ_{n,m} such that:

1. The group Γ_{n,m} is simple, finitely presented, torsion free and isomorphic to a free amalgam F ∗_{G} F where F, G are finitely generated free groups.

2. The group Γ_{n,m} has cohomological dimension 2.

3. Γ_{n,m} is automatic.

The question of existence of simple groups which are amalgams of free groups was raised by P.M. Neumann ([Neu], see also the Kourovka notebook [MK] problem 4.45). M. Bhattacharjee [Bha] constructed examples of amalgams of free groups which do not have any finite quotients. The groups Γ_{n,m} are constructed as lattices in Aut 𝔗_{n} × Aut 𝔗_{m} (where 𝔗_{k} denotes the k-regular tree). Considering the action of Γ_{n,m} on each of the trees 𝔗_{k}, k = n, m, we obtain two decompositions of Γ_{n,m} as amalgams A ∗_{C} B. The groups A, B and C, being torsion free lattices in Aut 𝔗_{k}, are free groups (note that [A : C] = [B : C] ∈ {n, m}). Moreover, using the superrigidity theorem 9 it follows that these are the only nontrivial decompositions of Γ_{n,m} as amalgamated products. This implies also that the groups Γ_{n,m} are mutually non isomorphic. These also form the first examples of finitely presented simple groups of finite cohomological dimension. We refer to [Sco] for a survey and discussion of various families of finitely presented simple groups constructed by R. Thompson, Higman, Brown and Scott. Geometrical description. When a subgroup D < Aut T_{1} × Aut T_{2} acts freely on T_{1} × T_{2}, it may be identified with the fundamental group of the quotient space Y = D\(T_{1} ×T_{2}). Note that Y is a square complex whose universal covering space is T_{1} × T_{2}. Square complexes whose universal covering space is a product of trees are characterized as those square complexes in which the link of every vertex is a complete bipartite graph. (Again under the analogy with semisimple groups consider the geometric characterization of locally symmetric spaces.) More generally, when the action is not free, D may be reconstructed as the fundamental group of a certain complex of groups ([Hae] and see [Ser] and [Bas] for the corresponding theory of graph of groups). This geometric way of considering subgroups and in particular cocompact lattices in Aut T_{1} × Aut T_{2} allows one to explicitly construct and modify such lattices. Note that any such finite square complex gives a finite presentation of its fundamental group. In [BM3] we give an explicit construction of the complexes associated with the groups Γ_{n,m} thus providing an explicit finite presentation of these torsion free simple groups. Minimal volumes. Kazhdan and Margulis [KM] have shown that for any semisimple Lie group there is a positive lower bound on the volume of Γ\G for any lattice Γ < G. In this vein let us mention the works of Lubotzky and Weigel, [Lu2], [LW], who determined the lattices of minimal covolume in the groups of the form SL_{2}(K), where K is a non archimedean local field.

In contrast, there is no lower bound for arbitrary lattices in either Aut T or Aut T_{1} × Aut T_{2}. Bass and Kulkarni, [BK], constructed cocompact lattices in Aut T, where T is a k-regular tree, k ≥ 3, with arbitrarily small covolume. I. Levitz determined precisely the (dense) set of (positive) rational numbers appearing as covolumes of uniform lattices in Aut T. Moreover, Bass and Lubotzky have shown in [BL2] that given any real number α > 0 there exists a non-uniform lattice Γ < Aut T such that Vol(Γ\Aut T) = α. Considering lattices in Aut T_{1} × Aut T_{2}, Y. Glasner has constructed examples of irreducible lattices Γ < Aut T_{1} × Aut T_{2} of arbitrarily small covolume. In these examples the subgroups H_{i} = \overline{pr_{i}(Γ)} < Aut T_{i} are not locally primitive.

However, a well known conjecture of Goldschmidt and Sims, translated into the language of lattices, asserts that for any given tree T there are only finitely many locally primitive lattices in Aut T. Thus in particular there is a lower bound on the covolume of such lattices. The Goldschmidt-Sims conjecture is usually stated as saying that for any n, m ≥ 3 there are only finitely many effective amalgams A ∗_{C} B of finite groups with [A : C] = n, [B : C] = m and C is maximal in both A and B. This was established by D. Goldschmidt [Gol] for the case n = m = 3. In view of the above results concerning the analogy between semisimple Lie groups and locally primitive groups of tree automorphisms, one is led to ask whether there is a positive lower bound on the covolume of lattices Γ < Aut T_{1} × Aut T_{2} having locally primitive projections (note that for reducible such lattices this would follow from the Goldschmidt-Sims conjecture). Studying this question Y. Glasner [Gla] has proved the following analog of the Goldschmidt-Sims conjecture:

Theorem 17. (Glasner) For any fixed primes p, q there are only finitely many effective complexes of groups consisting of a single square whose universal covering space is a product of regular trees 𝔗_{p} × 𝔗_{q} and whose fundamental group Γ < Aut 𝔗_{p} × Aut 𝔗_{q} is an irreducible lattice.

A central role in the proof of the above theorem is played by the Thompson-Wielandt theorem (Theorem 7). Recall that in establishing the lower bound on the covolume of a lattice in a semisimple Lie group G one may use ([KM], cf. [Rag]) the existence of a “Zassenhaus neighbourhood” U < G such that for every discrete subgroup Γ < G the elements of Γ ∩ U are contained in some connected nilpotent Lie subgroup of G. The Thompson-Wielandt theorem gives a neighbourhood whose intersection with locally primitive discrete groups is an l-group, and hence, in particular, nilpotent. Acknowledgements. I would like to thank Marc Burger, Yair Glasner, Alex Lubotzky and Bob Zimmer for many helpful discussions.

References

[AB] N. A’Campo and M. Burger. Réseaux arithmétiques et commensurateurs d’après G. A. Margulis. Invent. Math. 116(1994), 1–25.

[Bas] H. Bass. Covering theory for graphs of groups. J. Pure and Appl. Algebra 89(1993), 3–47.

[BK] H. Bass and R. Kulkarni. Uniform tree lattices. J. Amer. Math. Soc. 3(1990), 843–902.

[BL1] H. Bass and A. Lubotzky. Rigidity of group actions on locally finite trees. Proc. London Math. Soc. (3) 69(1994), 541–575.

[BL2] H. Bass and A. Lubotzky. Tree Lattices. A forthcoming book.

[Bha] M. Bhattacharjee. Constructing finitely presented infinite nearly simple groups. Comm. Algebra 22(1994), 4561–4589.

[Bor] A. Borel. Density properties for certain subgroups of semisimple groups without compact components. Annals of Math. 72(1960), 179–188.

[Bro] K.S. Brown. Buildings. Springer, 1989.

[Bur] M. Burger. Rigidity properties of group actions on CAT(0)-spaces. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 761–769, Basel, 1995. Birkhäuser.

[BM1] M. Burger and S. Mozes. CAT(-1)-spaces, divergence groups and their commensurators. J. Amer. Math. Soc. 9(1996), 57–93.

[BM2] M. Burger and S. Mozes. Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris Sér. I Math. 324(1997), 747–752.

[BM3] M. Burger and S. Mozes. Products of trees, Lattices and Simple Groups. In preparation.

[BMZ] M. Burger, S. Mozes, and R.J. Zimmer. Irreducible lattices in the automorphism group of a product of trees, Superrigidity and Arithmeticity. In preparation.

[Dan] S.G. Dani. A simple proof of Borel’s density theorem. Math. Z. 174(1980), 81–94.

[Fan] P.S. Fan. The Thompson-Wielandt Theorem. Proc. Amer. Math. Soc. 97(1986).

[FTN] A. Figa-Talamanca and C. Nebbia. Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, volume 162 of LMS. Cambridge University Press, 1991.

[Fur] H. Furstenberg. A note on Borel’s density theorem. Proc. Amer. Math. Soc. 55(1976), 209–212.

[Gla] Y. Glasner. Some Remarks on the Co-volume of Lattices Acting on a Product of Trees. Master’s thesis, Hebrew University, 1997.

[Gol] D.M. Goldschmidt. Automorphisms of trivalent graphs. Annals of Mathematics 111(1980), 377–406.

[Hae] A. Haefliger. Complexes of groups and orbihedra. In Group Theory from a Geometrical Viewpoint, pages 504–540. World Sci. Publishing, River Edge, NJ, 1990.

[KM] D. Kazhdan and G.A. Margulis. A proof of Selberg’s hypothesis. Math. Sbornik (1968).

[Liu] Y.S. Liu. Density of the commensurability group of uniform tree lattices. J. of Algebra 165(1994).

[Lu1] A. Lubotzky. Trees and discrete subgroups of Lie groups over local fields. Bull. Amer. Math. Soc. (N.S.) 20(1989), 27–30.

[Lu2] A. Lubotzky. Lattices of minimal covolume in SL_{2}: A nonarchimedian analogue of Siegel’s theorem μ ≥ π/21. J. Amer. Math. Soc. 3(1990), 961–975.

[Lu3] A. Lubotzky. Tree lattices and lattices in Lie groups. Combinatorial and Geometric Group Theory, Edinburgh 1993, LMS Lecture Notes Series 204, (1995), 217–232.

[LM] A. Lubotzky and S. Mozes. Asymptotic properties of unitary representations of tree automorphisms. In M.A. Picardello, editor, Harmonic Analysis and Discrete Potential Theory. Plenum Press, 1992.

[LMZ] A. Lubotzky, S. Mozes, and R.J. Zimmer. Superrigidity for the commensurability group of tree lattices. Comment. Math. Helv. 69(1994), 523–548.

[LW] A. Lubotzky and T. Weigel. Lattices of minimal covolume in SL_{2} over nonarchimedean fields. Proc. London Math. Soc. (To Appear).

[Mar] G.A. Margulis. Discrete subgroups of semisimple Lie groups, volume 17 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1991.

[MK] V. D. Mazurov and E. I. Khukhro, editors. Unsolved problems in group theory. The Kourovka notebook. Russian Academy of Sciences Siberian Division Institute of Mathematics, Novosibirsk, augmented edition, 1995.

[Moz] S. Mozes. On the congruence subgroup problem for tree lattices. In Proceedings of workshop on Ergodic theory and Lie groups. Tata Institute of Fundamental Research, Bombay, India, To appear.

[Neu] P.M. Neumann. The SQ-universality of some finitely presented groups. J. Austral. Math. Soc. 16(1973), 1–6. Collection of articles dedicated to the memory of Hanna Neumann, I.

[Rag] M. S. Raghunathan. Discrete Subgroups of Lie Groups. Springer-Verlag, Berlin Heidelberg New York, 1972.

[Sco] E.A. Scott. A tour around finitely presented infinite simple groups. In G. Baumslag and C.F. Miller III, editors, Algorithms and Classification in Combinatorial Group Theory, volume 23 of MSRI publications, pages 83–119. Springer-Verlag, 1989.

[Ser] J-P. Serre. Trees. Springer-Verlag, New York, 1980.

[Tho] J.G. Thompson. Bounds for orders of maximal subgroups. J. of Algebra 14(1970), 135–138.

[Tit] J. Tits. Sur le groupe des automorphismes d’un arbre. In A. Haefliger and R. Narasimhan, editors, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pages 188–211. Springer, New York, 1970.

[Wie] H. Wielandt. Subnormal subgroups and permutation groups. Lecture notes. Columbus, Ohio State Univ., 1971.

[Zim] R.J. Zimmer. Ergodic Theory and Semisimple Groups, volume 81 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 1984.

Shahar Mozes

Institute of Mathematics Hebrew University Jerusalem Israel mozes@math.huji.ac.il