Spring 2018

Abstract: Artin groups form a large class of groups including braid groups, free groups, and free abelian groups. They are related to Coxeter groups and have nice geometric properties. However many questions about the properties of Artin groups remain unanswered. In this talk, I will introduce the clique-cube complex, a CAT(0) cube complex constructed from a given Artin group, similar to the Deligne complex. I will discuss some of the properties of this cube complex, as well as how it can be used to show that a large class of Artin groups have trivial center and are acylindrically hyperbolic.

Abstract: I will discuss joint work with Rachel Skipper and Stefan Witzel, in which we prove that for any there exist simple groups of type but not . This result is new for all ; for example we now have the first known finitely presented simple groups that are not of type . Our examples in particular provide an infinite family of finitely presented simple groups that are pairwise non-quasi-isometric. The only previously known such family, due to Caprace-Rémy, consists of non-affine Kac-Moody groups over finite fields. Our examples arise from certain Nekrashevych groups, in the extended family of Thompson’s groups.

Abstract: An action of a finitely generated group G on a manifold M is called
“geometric” if it comes from an embedding of G as a lattice in a Lie group
acting transitively on M. In this talk, I’ll explain new joint work with
Maxime Wolff that characterizes geometric actions of surface groups on the
circle by dynamical rigidity. We show the only source of strong rigidity (in
a dynamical sense) for these actions is an underlying geometric structure:
from simple dynamical assumptions, we can in fact reconstruct a hyperbolic
surface.

Abstract: Given a finite-sheeted, possibly branched covering space between surfaces, it’s natural to ask how the mapping class group of the covering surface relates to the mapping class group of the base surface. In this talk, we will take a journey through this question for surfaces with boundary. It will feature appearances from the fundamental groupoid, the Birman-Hilden theorem, the Burau representation and new embeddings of the braid group in mapping class groups.

This is joint work with Alan McLeay

Abstract: In this talk we will define some Gromov hyperbolic simplicial complexes on which the group of outer automorphisms of a free group acts. This is in analogy to the very useful action of mapping class group on the curve complex. We will then focus on the outer automorphisms that act with positive translation length on these Out() complexes, specifically on the cyclic splitting complex. This is joint work with Derrick Wigglesworth.

Abstract: A large part of the success of geometric group theory can be traced to the Milnor-Schwartz Theorem, which states that when a group acts cocompactly on a space the large scale geometry of coincides with that of the Cayley graph of the group . Unfortunately, many interesting group actions are not cocompact, and require the introduction of additional uniformity conditions. I will discuss the example of the space of deformations of a non-compact Riemann surface, with the construction of a Thurston-like boundary for the corresponding Teichmüller space. This is joint work with Dragomir Šarić.

Abstract: A Kleinian group is convex cocompact if its orbit in hyperbolic 3-space is quasi-convex or, equivalently, that it acts cocompactly on the convex hull of its limit set in in hyperbolic 3-space.

Subgroup stability is a strong quasi-convexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasi-convexity condition above. Importantly, it coincides with quasi-convexity in hyperbolic groups and the notion of convex cocompactness in mapping class groups which was developed by Farb-Mosher, Kent-Leininger, and Hamenstädt.

Using the Morse boundary, I will describe an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups. Along the way I will discuss some known results about stable subgroups of various groups, including the mapping class group and right-angled Artin groups. The talk will include joint work with Matthew Gentry Durham and joint work with David Hume.

Abstract:TBA

Abstract:TBA