We consider exponential mixing for volume preserving random dynamical systems on surfaces. Suppose that (f_1,...,f_m) is a tuple of volume preserving diffeomorphisms of a closed surface M. We now consider the uniform Bernoulli random dynamical system that this tuple generates on M . We assume that this tuple satisfies a condition called being "expanding on average," which means that the log of the norm of unit tangent vectors grows in expectation, where the expectation is taken over all the realizations of the random dynamics. From this assumption we show quenched exponential mixing. (This is joint work with Dmitry Dolgopyat.)

I will discuss recent work with Minsky and Sisto, in which we prove that mapping class groups of finite-type surfaces---and more generally, colorable hierarchically hyperbolic groups (HHGs)---are asymptotically CAT(0). This is a simple but powerful non-positive curvature property introduced by Kar, roughly requiring that the CAT(0) inequality holds up to sublinear error in the size of the triangle. We use the asymptotically CAT(0) property to construct visual compactifications for colorable HHGs that provide Z-structures in the sense of Bestvina and Dranishnikov. It was previously unknown that mapping class groups are asymptotically CAT(0) and admit Z-structures. As an application, we prove that many HHGs satisfy the Farrell-Jones Conjecture, providing a new proof for mapping class groups (Bartels-Bestvina) and establishing the conjecture for extra-large type Artin groups. To construct asymptotically CAT(0) metrics, we show that every colorable HHG admits a manifold-like family of local approximations by CAT(0) cube complexes, where transition maps are cubical almost-isomorphisms.

Consider a closed surface M of genus greater than or equal to 2. For negatively curved metrics on M and their corresponding geodesic flow, we can study the topological entropy, the Liouville entropy, and the mean root curvature. In 2004, Manning showed that the topological entropy strictly decreases along the normalized Ricci flow if we start with a metric of variable negative curvature and asked whether monotonicity holds for the Liouville entropy. In this talk, we answer Manning's question and show that the Liouville entropy strictly increases along the flow. This talk is based on joint work with Butt, Humbert, and Mitsutani.

Artin groups and their related structures have long been of interest from both a geometric and topological perspective. I will give a brief overview of some recent results concerning quotients of Artin groups by powers of standard generators and their corresponding spaces. These quotients possess their own interesting and sometimes unintuitive geometry and topology, with some unexpected connections. We can leverage this divergence to prove new results for Artin groups themselves; as an example, we show that an Artin group which is simultaneously 2-dimensional, hyperbolic-type, and FC-type is fully residually hyperbolic and residually finite via a pseudo-Dehn filling procedure.

For a given group G, it is natural to ask whether one can classify all isometric G-actions on Gromov hyperbolic spaces. I will discuss a formalization of this problem based on the complexity theory of Borel equivalence relations. Our focus will be on actions of general type, i.e., non-elementary actions without fixed points at infinity, as these are particularly useful from the perspective of geometric group theory. The main result in this direction is the following dichotomy: for every countable group G, either all general type G-actions on hyperbolic spaces can be classified by an explicit invariant ranging in an infinitely dimensional projective space, or they are unclassifiable in a very strong sense. In terms of Borel complexity theory, we show that the equivalence relation associated with the classification problem is either smooth or K_σ-complete. Groups SL_2(F), where F is a countable field of characteristic 0, satisfy the former alternative, while non-elementary hyperbolic (and, more generally, acylindrically hyperbolic) groups satisfy the latter. The proof of the main theorem draws on several results of independent interest and provides new insights into the boundary dynamics of group actions on hyperbolic spaces. The talk is based on a joint paper with K. Oyakawa.

For $C^(l+)$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. In this talk I will give two the applications: Uniqueness of MME for finite horizon dispersing billiards and the robustly non-uniformly hyperbolic volume-preserving endomorphisms introduced by Andersson-Carrasco-Saghin. This is a joint work with Y. Lima and D. Obata.

Consider a space $X$ of non-positive curvature, with a group of isometries Gamma. Then Gamma naturally acts on various boundaries of X$, in particular the visual or geodesic boundary. When $X$ is a symmetric space of higher rank, e.g. SL(n,R)/SO(n) for n at least 3, these boundary actions become very rigid. I will discuss several instances of this such as local semi-rigidity in the grip of homeomorphisms as well as factors and other relations.

One constructs a fibered 3-manifold by thickening a surface by the interval and gluing its ends via a surface homeomorphism. In the classical finite-type setting, various forms of topological data of the gluing homeomorphism determine geometric information about the hyperbolic 3-manifold. Currently, there is a lot of research activity surrounding end-periodic homeomorphisms of infinite-type surfaces. As an "infinite type" analogue to work of Minsky in the finite-type setting, we show that given a subsurface Y of S, the subsurface projections between the "positive" and "negative" Handel-Miller laminations provide bounds for the geodesic length of the boundary of Y as it resides in the hyperbolic end-periodic mapping torus. In this talk, we'll discuss the motivating theory for finite-type surfaces and closed fibered hyperbolic 3-manifolds, show how these techniques are used in the infinite-type setting, and how the main theorems of this work return results to the closed, fibered setting.