From local to global similarity of matrix groups

Let G and H be groups of complex n×n matrices. We say that G is an H-like group if every matrix in G is similar to a matrix from H. For several groups H we consider two questions:

(A)

Is every H-like group (simultaneously) similar to a subgroup of H?

(B)

Is H the only H-like group containing H? Among other results we prove that the symmetric group S_n is the only S_n-like group containing S_n.

     

From local to global behavior in competitive Lotka-Volterra systems

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics.

The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point and the carrying simplex of the system lies to one side of its tangent hyperplane at , then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.

     

Two-generated groups acting on trees

We study 2-generated subgroups of groups that act on simplicial trees. We show that any generating pair {g,h}\{{g},h\} of such a subgroup is Nielsen-equivalent to a pair {f,s}\{f,s\} where either powers of f and s or powers of f and sfs^-1sfs^{-1} have a common fixed point if the subgroup ág,h?\langle {g},h\rangle is freely indecomposable. Analogous results are obtained for generating pairs of fundamental groups of graphs of groups. Some simple applications are given.     

Groups acting on circulant matrices

Summary If a groupG permutes a setI, andM is a multiplicative abelian group, a representation ofG onM ^I is given by permutation of coordinates. TheG-module homomorphisms intoM ^I arise from exponential maps. This framework encompasses those systems of functional equations that characterize generalized hyperbolic functions.     

A topological approach to the theory of groups acting on trees

Bass and Serre recast the foundations of combinatorial group theory in [7]. Here we apply the allied notions of fundamental group and covering space to redevelop their theory in a less combinatorial fashion; for example the Bass—Serre Structure Theorem is proved with no a priori knowledge of the group theoretic structure of the fundamental group of a graph of groups. Van Kampen's Theorem is used only once, in its simplest form (in the proof of Theorem 7). Cancellation arguments and normal form theorems, such as Britton's Lemma, are completely avoided; indeed they are incidental corollaries from our viewpoint. The tree which plays a central role in [7] appears in Theorem 2 as the natural analogue of the “strecken komplexe” introduced by A. Speiser [8], and subsequently also employed by R. Nevanlinna [5], to describe certain simply connected Riemann surfaces occurring in value distribution theory.     

Profinite completions of some groups acting on trees

We investigate the profinite completions of a certain family of groups acting on trees. It turns out that for some of the groups considered, the completions coincide with the closures of the groups in the full group of tree automorphisms. However, we introduce an infinite series of groups for which that is not so, and describe the kernels of natural homomorphisms of the profinite completions onto the aforementioned closures of respective groups.     

Locally compact groups acting on trees and propertyT

FollowingKazhdan, a separable locally compact groupG is said to have propertyT if the trivial representation is isolated in the dual space,, of equivalence classes of continuous irreducible unitary representations ofG. We generalize results ofMargulis—Tits by showing that groups which have propertyT can not be amalgams.Research supported by NSF.     

Groups acting on quasiconvex spaces and translation numbers

We prove that groups acting geometrically on -quasiconvex spaces contain no essential Baumslag-Solitar quotients as subgroups. This implies that they are translation discrete, meaning that the translation numbers of their nontorsion elements are bounded away from zero.

     

Dimension and randomness in groups acting on rooted trees

We explore the structure of the -adic automorphism group of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán.

We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups of generated by three random elements are full dimensional and that there exist finitely generated subgroups of arbitrary dimension. Specifically, our results solve an open problem of Shalev and answer a question of Sidki.

     

The automorphism tower of groups acting on rooted trees

The group of isometries of a rooted -ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in . This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group studied by R. Grigorchuk, and the group studied by N. Gupta and the second author.

In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as and . We describe this tower for all subgroups of which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of and .

More precisely, the tower of is infinite countable, and the terms of the tower are -groups. Quotients of successive terms are infinite elementary abelian -groups.

In contrast, the tower of has length , and its terms are -groups. We show that is an elementary abelian -group of countably infinite rank, while .

     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

Hecke algebras from groups acting on trees and HNN extensions

We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal $C^1$-completion. The paper unifies algebraic and analytic approaches by focusing on the common geometric thread. The results have implications for the general theory of totally disconnected locally compact groups.     

Groups acting transitively on compact CR manifolds of hypersurface type

Let be a compact homogeneous manifold with acting effectively and with a -invariant CR structure of hypersurface type; then any maximal compact subgroup acts transitively on .