Spring 2025

Abstract: Hyperbolic geometry has been a powerful tool in the study of manifold topology. Beyond the classical theory of surfaces, Thurston showed that the family of surface bundles over the circle is a rich source of hyperbolic 3-manifolds. In dimension 4, the correct analogue is given surface bundles over surfaces. In order for such a bundle to admit a hyperbolic metric, it needs to satisfy some conditions, such as being atoroidal and having signature zero. Surprisingly enough, the first examples of atoroidal surface bundles over surfaces were constructed only recently by Kent-Leininger. In this talk I’ll explain why these examples also have signature zero, meaning that they could admit hyperbolic metrics. This is joint work with J-F. Lafont and N. Miller.

Abstract:

Abstract:

Possible topics: asymptotic cones of groups, surface bundles over surfaces

It is a longstanding open question to determine which Artin groups are residually finite. Past results have followed from linearity (e.g., for spherical-type or virtually cocompact special Artin groups) or product decompositions in rank 3. We present a new approach to this problem using intermediate quotients to so-called Shephard groups. These Shephard groups possess their own interesting (and sometimes counterintuitive) geometry which we can leverage to give new information about their corresponding Artin groups in some cases. As a highlight of this connection, we show that an Artin group which is simultaneously 2-dimensional, hyperbolic-type, and FC-type is residually finite. One of the key features of the proof we will discuss is the fact that hyperbolic-type 2-dimensional Shephard groups are relatively hyperbolic, which is almost never true of Artin groups.

Abstract: The following elementary question seems to be open:

Suppose a group G of 2×2 matrices contains SL_2(Z), and is contained in SL_2(Z(\sqrt(2)), must G = SL_2(Z) or G = SL_2(Z[\sqrt(2)]) up to finite index?

This question fits in the general problem of understanding when a discrete subgroup in a (typically higher rank) Lie group has finite or infinite covolume. We will discuss some famous related problems, and prove some results based on the ideas of Margulis in the proof of arithmeticity and related work of Venkataramana-Chatterji.

All these notions will be explained, and the talk will be accessible (hopefully) to a large audience. Based on ongoing work with Subhadip Dey. 

Abstract: The growth rates of exponentially-growing groups have attracted attention since the observation by Koubi that hyperbolic groups have uniform exponential growth. Recent developments on the “growth spectrum” (i.e. the set of exponential growth rates of a same finitely generated group obtained from its different generating systems) have followed the progress of equation theory in groups. In particular, hyperbolic groups and certain acylindrically hyperbolic groups (including free-by-cyclic groups) have well-ordered growth spectrum. However, apart from a few cases (e.g. free groups), little is known about the minimizing set of generators. In fact, examples of explicit growth rate calculations are still very sparse. After surveying these topics, I will detail some estimates for growth rates of free-by-cyclic groups.

Abstract: In this talk I will survey a number of recent results regarding (relative) profinite rigidity of certain groups (3-manifold groups,Coxeter groups, free-by-cyclic groups, Kaehler groups). Here profinite rigidity asks how much of information about a finitely generated residually finite group can be recovered from its finite quotients. From an algebraic geometry viewpoint this is essentially asking when the algebraic fundamental group determines an aspherical projective variety up to biholomorphism (assuming residual finiteness of the topological fundamental group). Much of the input will come from developments around the world of 3-manifold topology, building on the Virtual Fibring Theorem of Agol. With this in hand (and time permitting) I will discuss work of Wilton—Zalesskii, Wilkes, and Liu on rigidity amongst 3-manifold groups, work of myself and Kudlinska on rigidity amongst free-by-cyclic groups, and work of myself, Llosa Isenrich, Py, Stover, and Vidussi on rigidity amongst Kaehler groups.

Abstract:
Every finitely generated group G has an associated topological space, called a Morse boundary, that captures the hyperbolic-like behavior of G at infinity. It was introduced by Cordes generalizing the contracting boundary invented by Charney–Sultan. In this talk, we study subgroups arising from connected components in Morse boundaries of right-angled Coxeter groups and of such that are quasi-isometric to right-angled Coxeter groups. This talk is based on two projects. One is joint work with Bobby Miraftab and Stefanie Zbinden. The other one is joint work in progress with Matthew Cordes and Kim Ruane.

Abstract:

Abstract:

Abstract:

Abstract:

Abstract: