Texas A&M University, Department of Mathematics

Special Year on

Asymptotic Group Invariants and their Applications

Spring Workshop 1, February 26, 2005

Schedule

317 Milner Hall, Texas A&M University, College Station TX
Organizers: Rostislav Grigorchuk, Gilles Pisier and Zoran Sunik of Texas A&M University

Abstracts

Volodymyr Nekrashevych of International University Bremen, Germany
Teichmüller space and the "twisted rabbit" question of John Hubbard

We will give a solution of the "twisted rabbit" question of J. Hubbard, which uses the Teichmüller theory. We will show that the pull-back map on the Teichmüller space of a punctured sphere, induced by a class of branched coverings of the plane, is projected onto a rational map of the moduli space. Iterations of this rational map can be used to solve the "twisted rabbit" question. A connection with random iterations will also be discussed.

Kevin Pilgrim of Indiana University
Self-similar groups and conformal dynamics

Let G be a finitely generated subgroup of the automorphism group of an infinite rooted tree with constant d-fold branching. Under certain conditions (when G is selfsimilar, contracting, and recurrent) there is a naturally associated topological dynamical system f: X --> X. Work of V. Nekrashevych identifies X as a boundary at infinity of a Gromov hyperbolic 1-complex.  As a consequence, X inherits a preferred quasisymmetry class of metrics.  By establishing strong finiteness properties analogous to those enjoyed by Gromov hyperbolic groups, we show that in these metrics, f: X --> X is conformal in the sense that iterates distort the roundness of balls by at most a constant factor.  As applications, we obtain new numerical topological invariants and new rigidity theorems for certain expansive dynamical systems. This is joint work with P. Haissinsky, Univ. de Provence.

Said Sidki of Universidade de Brasília, Brazil
Endomorphisms of the finitary group of isometries of the binary tree

Let T be the binary tree, A its group of isometries, F the subgroup of finite-state isometies and G the subgroup of finitary isometries. The group G is a locally finite dense subgroup of A.  In a joint paper with A. Brunner (J. Algebra, 1997) we studied in detail the structure of G and established the form of endomorphisms of G induced by conjugation by elements of A and those by elements of F. Recognizing when such endomorphisms are automorphism has proven to be very difficult. We also showed that N_A(G) contains a copy of A itself.  I will talk about our recent joint work where we continue the study of this recognition problem and further construct in N_F(G) a free product of two elementary abelian groups each of countably infinite rank.