ICM 1998, Berlin, Aug. 18–27

Abstracts of Plenary and Invited Lectures

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 20E06, 20E08, 20E32, 20F32, 22D40, 22E40 Mozes, Shahar, Institute of Mathematics, Hebrew University, Jerusalem, Israel Products of trees, lattices and simple groups

The group of automorphisms of a locally finite tree, denoted Aut T, is a locally compact group which exhibits behaviour 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 T_{i}, i = 1, 2 is a (bi-)regular tree. The results obtained concerning the structure of lattices in Aut T_{1} × Aut T_{2} enable us to show the existence and construct the first examples of finitely presented torsion free simple groups, see [1]. A lattice Γ < Aut T_{1} × Aut T_{2} is reducible if its projections on each factor Aut T_{i} are discrete. Denoting the closures of the projections of Γ by H_{i} = \overline{pr_{i}(Γ)}, i = 1, 2, we observe that for cocompact lattices H_{i} ≠ Aut T_{i}. We show that imposing various “largeness” conditions on the groups H_{i}, i = 1, 2 allows one to obtain a rich structure theory for these lattices. A (closed) subgroup H < Aut T will be called locally primitive if for every vertex x ∈ T its stabilizer in H acts as a primitive permutation group on the set of adjacent edges. The following result may be viewed as an “arithmeticity theorem”. Theorem 1, [2]. Let Γ < Aut T_{1} × Aut T_{2} be an irreducible cocompact lattice. Let H_{i} = \overline{pr_{i}(Γ)} be the closures of the projections, i = 1, 2. Assume that each H_{i} < 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 yields a continuous surjection from H_{1} × H_{2} onto a semisimple Lie group over some local fields and the image of 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)). A related result is a geometric rigidity theorem asserting that if Γ < Aut (T_{1}) × Aut (T_{2}) is a cocompact lattice with locally primitive projections and Γ is isomorphic to a lattice Γ′ < Aut (T′_{1}) × Aut (T′_{2}) then the isomorphism is induced by an isometry between T_{1} × T_{2} and T′_{1} × T′_{2}. Further results, jointly with Marc Burger, study the normal subgroups structure of irreducible lattices in Aut T_{1} × Aut T_{2}. Again one needs to put conditions on the “size” of the projections of the lattice. For a locally compact group H let us denote by H^{(∞)} the intersection of all its cocompact closed normal subgroups. Assuming that the closures of the projections of an irreducible lattice act transitively on the boundary of the tree we have the following analog of Margulis’ normal subgroup theorem: Theorem 2, [1]. Let Γ < Aut T_{1} × Aut T_{2} be a cocompact lattice such that H_{i} := \overline{pr_{i}(Γ)} acts transitively on ∂T_{i} and H_{i}^{(∞)} is of finite index in H_{i}, i = 1, 2. Then, any nontrivial normal subgroup of Γ is of finite index. Unlike finitely generated linear groups which are necessarily residually finite, lattices in Aut T_{1} × Aut T_{2} may be non residually finite. We establish a criteria for ensuring that certain lattices are not residually finite. Using this criteria together with theorem 2 we show Theorem 3, [1]. For every pair (n, m) of sufficiently large even integers there exists a finite square complex Y_{n,m} whose universal covering is the product T_{n} × T_{m} of regular trees of respective degrees n and m and whose fundamental group Γ_{n,m} (which is a cocompact lattice in Aut T_{n} × Aut T_{m}) satisfies:

(1) The group Γ_{n,m} is simple, finitely presented and torsion free.

(2) Γ_{n,m} ≡ F ∗_{G} F where F, G are finitely generated free groups, [F : G] ∈ {n, m}.

(3) If Γ_{n,m} is isomorphic to Γ_{k,l}, then the corresponding complexes Y_{n,m} and Y_{k,l} are isomorphic, and {n, m} = {k, l}.

(4) The cohomological dimension of Γ_{n,m} is 2.

(5) Γ_{n,m} is an automatic group. We remark that the question whether amalgams F ∗_{G} L of free groups can be simple was raised by P. M. Neumann [4], [3].

References

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

[2] M. Burger, S. Mozes and R. J. Zimmer, Irreducible lattices in the automorphism group of a product of trees, Superrigidity and Arithmeticity, In preparation, [3] V. D. Mazurov, E. I. Khukhro, Eds., Unsolved Problems in Group

Theory. The Kourovka Notebook. Novosibirsk, 1992

[4] P. M. Neumann, The SQ-universality of some finitely presented groups, J. Austr. Math. Soc. XVI (1973) 1–6