Texas A&M University, Department of Mathematics

Special Day on

Groups and Dynamics

Fall Workshop, November 18, 2006

Schedule

317 Milner Hall, Texas A&M University, College Station TX Organizers: Rostislav Grigorchuk, Volodymyr Nekrashevych and Zoran Šunić of Texas A&M University

Abstracts

Tullio Ceccherini-Silberstein of Universitá del Sannio, Benevento, Italy
Minimal topological actions do not determine the measurable orbit equivalence class
Joint work with G. Elek

We construct a minimal topological action F of a non-amenable group on a compact space X which is amenable but non-uniquely ergodic: there exist two ergodic invariant measures m and M such that (F,X,m) and (F,X,M) are not orbit equivalent measurable equivalence relations.


Alexander I. Bufetov of Rice University
Existence and uniqueness of the measure of maximal entropy for the Teichmueller flow on the moduli space of abelian differentials
Joint work B.M. Gurevich

The moduli space of abelian differentials carries a natural Lebesgue measure class, and, by the Theorem of H.Masur and W.Veech, the Teichmueller flow on the moduli space of abelian differentials preserves a finite ergodic measure in  the Lebesgue measure class. The entropy of the flow with respect to the absolutely continuous measure has been computed by Veech in 1986.

The main result of this talk, obtained by B.M. Gurevich and the speaker, is that the absolutely continuous measure is the unique measure of maximal entropy for the Teichmueller flow.

The first step of the proof is an observation that the absolutely continuous measure has the Margulis property of uniform expansion on unstable leaves. After that, the argument proceeds in Veech's space of zippered rectangles. The flow is represented as a symbolic flow over a countable topological Bernoulli chain and with a Hoelder roof function depending only on the future. Following the method of Gurevich, the flow is then approximated by a sequence of flows whose suspension functions depend on only one coordinate in the sequence space. For these, conditions for existence and uniqueness of the measure of maximal entropy are known by theorems of Gurevich and Savchenko. Since the roof function of our initial flow is Hoelder, the approximation is rapid enough and yields maximality of entropy for the smooth measure as well as the uniqueness of the measure of  maximal entropy.


Rostislav Grigorchuk of Texas A&M University
Homomorphic images of branch groups and Serre Property (FA)
Joint work with T. Delzant.

It is shown that a finitely generated branch group has Serre Property (FA) if and only if it does not surject onto an infinite cyclic or infinite dihedral group. Examples of finitely generated self-similar branch groups surjecting onto the infinite cyclic or the infinite dihedral group are constructed.


Volodymyr Nekrashevych of Texas A&M University
C*-algebras and holomorphic dynamics

We will talk about Cuntz-Pimsner algebras associated with hyperbolic rational functions: the Cuntz-Pimsner algebras of the iterated monodromy groups and the cross-product algebras of the action of the rational function on its Julia set. We will discuss their basic properties and compute their K-theory. This will imply that many of such algebras are isomorphic to each other. On the other hand, we will see how one can reconstruct the Julia set of a rational function from the gauge action on the Cuntz-Pimsner algebra of the iterated monodromy group.