The finite index subgroups of a finitely presented group generate a topology on the group. We will discuss using examples how this relates to the organization of a group's finite quotients, and introduce the ideas of profinite rigidity and flexibility.
The Torelli group of a surface is a natural yet mysterious subgroup of the mapping class group. We will discuss a few recent results about finiteness properties of the Torelli group, as well as a result about the cohomological dimension of the Johnson filtration.
The fundamental groups of knot complements have lots of finite quotients. We give a criterion for a hyperbolic knot in the three-sphere to be distinguished (up to isotopy and mirroring) from every other knot in the three-sphere by the set of finite quotients of its fundamental group, and we use this criterion as well as recent work of Baldwin-Sivek to show that there are infinitely many hyperbolic knots distinguished (up to isotopy and mirroring) by finite quotients.
The braid group (which encodes the braiding of n strands) has a canonical projection to the symmetric group (recording where the ends of the strands go). We ask the question: what are the extensions of the symmetric group by abelian groups that arise as quotients of the braid group, by a refinement of this canonical projection? To answer this question, we study a particular twisted coefficient system for the symmetric group, called the integral pair module. In this module, we find the maximal submodule in each commensurability class.
In the 20th century, Thurston proved two classification theorems, one for surface homeomorphisms and one for branched covers of surfaces. While the theorems have long been understood to be analogous, we will present new work with Belk and Winarski showing that the two theorems are in fact special cases of one Ubertheorem. We will also discuss joint work with Belk, Lanier, Strenner, Taylor, Winarski, and Yurttas on algorithmic aspects of Thurston’s theorem. This talk is meant to be accessible to a wide audience.
