We give an orbit classification theorem and a general structure theorem for actions of groups of homeomorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. As applications, we will discuss some classification results about Homeo(M) and Diff(M) acting on N when dim(N)-dim(M) is not too big. This is joint work with Kathryn Mann.

In this talk, I'll present a strong form of symbolic coding for the geodesic flow on a compact, CAT(-1) space. A coding of this sort for negatively-curved Riemannian manifolds goes back to Bowen and has a wide range of implications for the dynamics of geodesic flows. We obtain the same results on CAT(-1) spaces, using machinery developed by Bowen and Pollicott. Our work also handles geodesic flow for projective Anosov representations.

There is a rich connection between homogeneous dynamics and number theory. Often in such applications it is desirable for dynamical results to be effective (i.e. the rate of convergence for dynamical phenomena are known). In talk, I will discuss the effective equidistribution of horospherical flows on quotients of SL(n,R) by a lattice and use this to prove an effective rate for the equidistribution of arithmetic progressions in abelian horospherical flows. I will then describe an application to studying the distribution of almost-prime times in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture.

Behrstock-Hagen-Sisto have shown how the "hulls" of finite sets constructed by Behrstock-Kleiner-Minsky-Mosher for mapping class groups can be approximated by CAT(0) cube complexes of controlled dimension. In work with Durham and Sisto, and using the machinery developed by Bestvina-Bromberg-Fujiwara-Sisto, we show that these approximations can be made stable, meaning that a bounded change in the data yields cube complexes that differ by a bounded number of hyperplane deletions and additions. As applications we define approximate equivariant barycenters for finite sets, and construct bicombings of mapping class groups. The proof works in the context of "colorable hierarchically hyperbolic spaces".

An immediate corollary of Nielsen-Thurston classification of surface homeomorphisms is that if two surface homeomorphisms f and g have homeomorphic mapping tori, then f is pseudo-Anosov if and only if g is too. Using hyperbolization theorem and rigidity results, the hypothesis can be weakened to quasi-isometric mapping tori. We show the analogous result for free group automorphisms: if two free group automorphisms have isomorphic mapping tori, then one is fully irreducible and atoroidal if and only if the other is fully irreducible and atoroidal. It is unknown if the hypothesis can be weakend quasi-isometric mapping tori.

Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery--Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery--Zippin) that it suffices to rule out the case of the additive group of $p$-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group.

A group is said to be of type $F_n$ if it admits a classifying space with compact $n$-skeleton. We will consider the class of R\"{o}ver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type $F_{n-1}$ but not $F_n$ for each $n$. These are the first known examples for $n\geq 3$. As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups.

This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky

We prove that for the harmonic measure associated to a random walk on Out(F) satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This is joint work with Ilya Kapovich, Joseph Maher, and Catherine Pfaff.

Mapping class groups inherit a natural topology from the compact-open topology on homeomorphism groups. When the underlying surface is of infinite type, this topology is no longer discrete; however, in this case, mapping class groups remain Polish. We will explain how to see that mapping class groups are Polish groups and some applications.

This is joint work with Juliette Bavard. The curve graph for an infinite type surface is degenerate, in that it has diameter two, so many facts about curve graphs of finite type surfaces fail to generalize. The loop graph, which is infinite diameter and hyperbolic, is a possible replacement. We give a concrete description of the Gromov boundary of the loop graph as the space of cliques of high-filling rays. We then study the action of the mapping class group of the surface on this boundary. As an application, we give a condition certifying two loxodromic elements as having anti-aligned axes, which via a theorem of Bestvina-Fujiwara gives many quasimorphisms on the mapping class group.