In this talk, I will discuss the use of Freudenthal compactifications of infinite-type surfaces and their uncountable fundamental groups to study their mapping class groups. In particular, when S is an orientable infinite type surface S with no planar ends and without boundary, the (extended) mapping class group of S is isomorphic to the mapping class group of the Freudenthal compactification of S. We show that such compactifications are the quotient of a disk by a countable edge-pairing on the boundary circle and thus may be constructed by attaching a single 2-cell to a one-dimensional Peano continuum. This provides an avenue for applying established technology for fundamental groups of one-dimensional Peano continua to big mapping class groups. This work is joint with Jesús Hernández Hernández and Curtis Kent.

Local product structure of a measure has been used as a key ingredient for showing various mixing properties of dynamical systems. One prominent example of this is Ornstein and Weiss's proof of the Bernoulli property for geodesic flows in negative curvature. I will discuss recent work establishing local product structure using a nonuniform Gibbs property for a class of equilibrium states for the geodesic flow for some systems beyond negative curvature. This is joint work with Dave Constantine, Alena Erchenko, Noelle Sawyer, and Grace Work.

The truncated braid groups are natural group theoretic constructions which turn out to have quite interesting topological and geometric implications. In this talk, I will explore two phenomena that show up with these groups. First, I will explain a topological perspective on a theorem of Coxeter that connects truncated braid groups with Platonic solids. And second, I will discuss a connection between these groups and the Burnside groups of combinatorial group theory.

Dehn surgery is a classical operation in which one converts one three-manifold to another by first removing a solid torus, and then gluing it back in in a different way. The first operation is called "drilling" and the second "filling". Both of these operations have group-theoretic interpretations in the world of hyperbolic and relatively hyperbolic groups. I will explain those interpretations and applications related to the Cannon conjecture about groups acting on the two-sphere. This talk is based on joint work with Groves, Haïssinsky, Osajda, Sisto, and Walsh.

A sequence of groups or spaces \(X_n\) exhibits homological stability if for every \(i\), their \(i\)-th homology group \(H_i(X_n)\) is independent of \(n\) for large enough \(n\). This phenomenon occurs in a wide variety of groups or spaces, such as the symmetric groups, general linear groups, symplectic groups, mapping class groups, configuration spaces, and many more. Homological stability even extends to many twisted coefficient systems, especially those that have polynomial growth. Most theorems have a range that depends on the degree of the polynomial growth. In this talk, I will give examples whose the range is uniform, i.e. independent of the degree.

These examples are important for applications to arithmetic statistics, such as the asymptotics of moments of \(L\)-functions, as well as for computing the stable homology of the Torelli subgroups of the mapping class groups.

Farb initiated a program to study special maps between certain moduli spaces. The goal of the program is to characterize the maps in terms of the complex structure of the associated varieties. In this setting we introduce a conjecture concerning the space of polynomials, and discuss it's connections to Teichmüller theory and braid groups.

Classically, the dynamics of geodesic and frame flows were studied for compact (and more generally, for finite volume) hyperbolic manifolds, and heavily relied on spectral gaps and representation theory. In recent decades, there have been many developments for infinite volume hyperbolic manifolds, in particular for exponential mixing. In this setting, It turns out that representation theory has limitations and Dolgopyat-type techniques are advantageous. Having established exponential mixing, it is then interesting to go in the reverse direction and look at spectral gaps and representation theoretic results. I plan to talk about some older work and a recent joint work with Dubi Kelmer and Osama Khalil to confirm a conjecture of Mohammadi--Oh.

For a group \( G \) acting on a regular rooted \(d\)-ary tree \(T_d\) and on its boundary \(\partial T_d\) we consider the diagonal actions of \(G\) on the powers of \(T_d\) and \(\partial T_d\). For the action of the full group \(\mathrm{Aut}(T_d)\) of automorphisms of \(T_d\) we describe the ergodic decomposition of its action on \((\partial T_d)^n\) for all \(n\geq 1\). To achieve it we analyze the orbits of \(n\)-tuples of elements of vertices of any fixed finite level of \(T_d\). For a subgroup \(G\) of \(\mathrm{Aut}(T_d)\) the corresponding orbits may be smaller, but sometimes they coincide with the orbits of the full group of automorphisms for all levels. In this case we say that the action of \(G\) on \(\mathrm{Aut}(T_d)\) is maximally tree \(n\)-transitive. For example, maximal tree 1-transitivity is equivalent to level transitivity of the action of \(G\) on \(T_d\). It follows from the results of Bartholdi and Grigorchuk that Grigorchuk group and Basilica group act maximally tree 2-transitively on \(\partial T_2\). We show that the action of Grigorchuk group on \(\partial T_2\) is, in-fact, maximally tree 4-transitive but none of Grigorchuk groups \(G_{\omega}\) is maximally 5-transitive. The talk is based on a joint work with Rostislav Grigorchuk and Zoran Sunic.

Commensurating actions govern how a group can act on non-positively curved cube complexes. We obtain a complete picture of them for a class of finitely generated groups acting on rooted trees: contracting self-similar branch groups. The main application is a proof of Property FW for the iterated monodromy group of the subdivision rule generating the classical square Sierpiński carpet. Property FW can be viewed as a natural combinatorial weakening of Property (T). The carpet group the first example of an infinite finitely generated amenable group with Property FW. Joint with Nicolas Matte Bon and Volodia Nekrashevych.