Carolyn Abbott (Brandeis University)

Math & Stats Department Colloquium

Friday, February 10th, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Carolyn Abbott (Brandeis University)

Title: Big mapping class groups and their actions on hyperbolic graphs

Abstract: Given a surface with nite genus and nitely many punctures, there are two important objects naturally associated to it: a group, called the mapping class group, and an innite-diameter hyperbolic graph, called the curve graph. The mapping class group acts by isometries on the curve graph, and this action has been extremely useful in understanding the algebraic and geometric properties of mapping class groups. Nielsen and Thurston give a powerful classication of elements of a mapping class group, in which the most interesting and complex elements correspond to those with the dynamically richest actions on the curve graph. When the surface has innite genus or innitely many punctures, the mapping class group is much more complicated, and this classication no longer holds. In this talk, I will explain where the complications arise in this innite-type setting, focusing on the action of a mapping class group on an associated hyperbolic graph generalizing the curve graph. I will describe several constructions of elements which have dynamically rich actions on this graph which do not appear in the nite-type setting. This is joint work with Nick Miller and Priyam Patel.

Dr. Abbott is an Assistant Professor in the Mathematics Department at Brandeis University. Previously, she was an NSF Postdoctoral Fellow at Columbia University (2019-2021), an NSF Postdoctoral Fellow at UC Berkeley (2018-2019), and a Morrey Visiting Assistant Professor, also at UC Berkeley (2017-2018). She received her Ph.D. from the University of Wisconsin-Madison in 2017, where she studied geometric group theory under the supervision of Tullia Dymarz.

Dr. Abbott's research interests include geometric group theory and low-dimensional topology. In particular, she is interested in group actions by isometries on hyperbolic spaces, especially acylindrical actions. The kinds of groups she thinks about include hyperbolic and relatively hyperbolic groups, mapping class groups, Out(Fn), CAT(0) groups, three manifold groups, hierarchically hyperbolic groups, and many more.