Abstract: We'll connect certain types of braided groups of homeomorphisms of Cantor sets with subgroups of some big mapping class groups studied by Funar-Kapoudjian and Aramayona-Funar. We'll sketch some tools that can be used to study these groups and show how they relate to some subcomplexes of the curve complex.

Abstract: Let \Gamma be a Zariski dense Kleinian Schottky subgroup of PSL_2(\mathbb{C}). Let \Lambda_{\Gamma}\subset \mathbb{C} be its limit set, endowed with a Patterson-Sullivan measure \mu. I will present the joint work with Jialun Li and Frédéric Naud in which we showed that the Fourier transform of \mu enjoys polynomial decay. As a corollary, all limit sets of Zariski dense Kleinian groups have a positive Fourier dimension. This is a PSL_2(\mathbb{C}) version of the PSL_2(\mathbb{R}) result of Bourgain-Dyatolv and uses the decay of exponential sums based on Bourgain-Gamburd sum-product estimate on \mathbb{C}. These bounds on exponential sums require a delicate non-concentration hypothesis. We verify the condition using some representation theory; we realize the Patterson-Sullivan measure as the stationary measure of certain random walks on PSL_2(\mathbb{C}) with the finite exponential moment, which is of independent interest. I will outline the main ingredients of the proof and talk about some future problems.

Abstract: The stable commutator length function measures the growth rate of the commutator length of powers of elements in the commutator subgroup of a group. In this talk, we will discuss the stable commutator length function on the mapping class groups of infinite-type surfaces which satisfy a certain topological characterization. In particular, we will show that stable commutator length is a continuous function on these big mapping class groups, as well as that the commutator subgroups of these big mapping class groups are both open and closed. Along the way to proving our main results, we will discuss certain topological properties of a class of infinite-type surfaces and their end spaces which may be of independent interest. This talk represents joint work with Priyam Patel and Alexander Rasmussen.

Abstract: A classical result of Khintchine’s provides a zero-one law for the Lebesgue measure of points in Euclidean space with a given quality of approximation by rational points. In 1984, Mahler asked whether a similar law holds for Cantor’s middle thirds set. His question is part of a long history of results and conjectures aiming at showing that unlikely intersections between Diophantine sets and natural subsets of Euclidean space only occur for well-understood algebraic reasons. Some of these elementary Diophantine questions ultimately lead to difficult problems at the interface of homogeneous dynamics and spectral theory of automorphic forms. I will describe recent joint work with Manuel Luethi leading to progress on Mahler’s problem by linking it to a notion of “sparse Hecke operators”. The talk will serve as an overview of this area and no background in the aforementioned topics will be assumed.

Classifying the actions of a given group on different hyperbolic metric spaces is a natural but typically very difficult problem. Recently, several authors have classified the hyperbolic actions of several families of classically-studied metabelian groups. In this talk we will describe how commutative algebra may be used as a tool to approach these classification problems in a uniform way, and extend the classification to larger families of metabelian groups. This is joint work with Carolyn Abbott and Sahana Balasubramanya.

Abelianizations of mapping class groups of finite-type surfaces are known by the work of Powell and Harer to always have either trivial or "small" abelianizations. We show that this is not the case for infinite-type surfaces. In fact, some of these abelianizations contain copies of uncountable direct sums of Qs. In this talk we will survey what is currently known for these groups, including how our results fit into the current state of affairs, and offer several natural questions. We will also give a brief sketch as to how we find nontrivial elements in the abelianizations.

TBA