Texas A&M University, Department of Mathematics

Special Year on

Asymptotic Group Invariants and their Applications

Fall Workshop 2, December 8, 2004

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

Tullio Ceccherini-Silberstein of Universitá del Sannio, Benevento, Italy
Automata, linear languages and their growth

A language is a subset of words over a finite alphabet. There are several classes of languages: for instance a "regular" language is a language recognized by an "automaton"; "context-free" languages are the languages generated by "context-free grammars", etc. We first show that slightly modifying the notion of an automaton (yielding the definition of a "bilateral" automaton) we can characterize the "linear" languages (a subclass of context-free languages containing the regular languages) as those recognized by such bilateral automata. We then present an algorithm which determines whether a given linear language has "polynomial" or "exponential" growth. Finally we show that "ergodic" linear languages are "growth-sensitive": ergodic means that the bilateral automaton is strongly connected (as an oriented graph) and growth-sensitivity means that g(L') < g(L) where g is the growth rate and L' is a sublanguage of L obtained by forbidding some non-trivial (sub)word in L.

Gabriel Dos Reis of Texas A&M University
Application of loop groups to constant mean curvature surfaces

Understanding the geometry and structure of surfaces with nonzero constant mean curvature (CMC surfaces) is a long standing puzzling problem for Geometers. That problem has seen a renewed interest in the last two decades with (a) the positive answer of H. Wente to H. Hopf's question about the existence of compact CMC surfaces of positive genus; and (b) the recent work of J. Dorfmeister, F. Pedit and H. Wu on the Weierstrass-type representation of harmonic maps in symmetric spaces. This talk will first review the so-called DPW method, then explore practical issues related to loop group factorizations, numerical constructions of CMC surfaces and relations to their extrinsic geometries.

Alexander Fel'shtyn of Universität Siegen, Germany
Reidemeister number of automorphisms of Gromov hyperbolic groups and Baumslag-Solitar groups

In the article "The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion" (K-theory 8 (1994) no.4, 367-393, A. L. Fel'shtyn and R. Hill) we  conjectured that if a group is finitely generated, has exponential growth and the group endomorphism is injective, then the number of twisted conjugacy classes is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups (Levitt-Lustig, Fel'shtyn) and for the family of Baumslag-Solitar groups (Fel'shtyn-Goncalves).  It was shown by Goncalves and Wong that  the conjecture does not hold in general.

Rostislav Grigorchuk of Texas A&M University
The Ihara zeta function for infinite groups and graphs
 
We will define the Ihara zeta function for infinite regular graphs and give nontrivial examples and computations. The problem of computation of this function will be related to the spectral theory of graphs and groups (in particular to the computation of Kesten - von Neumann - Serre spectral measure), which in turn will be related to the question of integrability of multidimensional rational maps whose dynamics will also be in our focus.

Natasha Macura of Trinity University
Quasi-isometry classification of the mapping tori of automorphisms of finitely generated free groups

The full quasi-isometry classification of the mapping tori of automorphisms of finitely generated free groups is still an open problem where one can ask a number of interesting and challenging questions. We will prove that the mapping tori of two automorphisms of finitely generated free groups that grow polynomially with different degrees cannot be quasi-isometric and discuss several different routes toward a more complete quasi-isometry classification.

Bogdan Petrenko of Texas A&M University
On pairs of matrices generating matrix rings

We will discuss some consequences of a lemma of W. Burnside.