Spring 2019 GGD WORKSHOP

Titles and Abstracts

Darren Creutz

Title: Relative Entropy and an Entropy Gap for Some Groups Without (T)

Abstract: Furstenberg introduced entropy for stationary (not necessarily measure-preserving) actions of groups which Kaimanavich and Vershik used in a crucial way to show that a group is amenable iff it admits a trivial Poisson Boundary.

Nevo showed that for groups with (T) there is a strong opposite condition: every stationary of a (T) group is either measure-preserving or has entropy at least some positive constant (independent of the action).

I will present a generalization of Furstenberg entropy to maps between stationary spaces and from this, show that certain product groups not having (T) also have an entropy gap for their actions.

Kasia Jankiewicz

Title: Cubical dimension of groups and uniform exponential growth

Abstract: The cubical dimension of G is the infimum n such that G admits a proper action on an n-dimensional CAT(0) cube complex. For each n, we construct examples of small cancellation groups with cubical dimension bounded below by n. This is related to the question of uniform exponential growth of cubulated groups.

Dan Thompson

Title: Symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows

Abstract: The geodesic flow on a compact locally CAT(-1) metric space, first studied by Gromov, is a far-reaching generalization of the geodesic flow on a closed negative curvature Riemannian manifold. While one expects these flows to exhibit similar behaviour to the classical case, the lack of smooth structure has been a major obstacle to extending many of the finer aspects of the dynamical theory to this setting. Our new approach to this problem is to show that such geodesic flows are metric Anosov flows.

A metric Anosov flow, or Smale flow, is a topological flow equipped with a continuous bracket operation which is an abstraction of the local product structure from uniform hyperbolicity. In 1987, Pollicott showed that a version of Bowen's construction of symbolic dynamics for Axiom A flows can be extended to this setting. By symbolic dynamics, we mean there exists a suspension flow over a shift of finite type which describes the original dynamics. By taking additional care in the construction, we are able to verify that the roof function and factor map can be taken to be Lipschitz in our setting. This is achieved by using carefully chosen geometric rectangles as the building blocks for the construction.

With this additional ingredient, the symbolic dynamics machine switches on and ergodic-theoretic results which are true for Axiom A flows are extended to this setting. For example, we obtain that the Bowen-Margulis measure for the geodesic flow is Bernoulli and satisfies the Central Limit Theorem. Our techniques also extend to the geodesic flow associated to a projective Anosov representation, which verifies that the full power of symbolic dynamics is available in that setting. This is joint work with Dave Constantine and Jean-Francois Lafont.

Kurt Vinhage

Title: Smooth Rigidity for Cartan Actions of Higher-Rank Abelian Groups

Abstract: We will discuss recent progress on the Katok-Spatzier conjecture: any uniformly hyperbolic $\R^k$ or $\Z^k$ action without rank-one factors is smoothly conjugate to an algebraic one. One of the main difficulties is producing a homogeneous space structure from the dynamical assumptions. We develop new tools coming from the theory of topological groups to build such group actions and homogeneous structures. Joint with Ralf Spatzier.