Dehn function, motivated by the isoperimetric function in Riemannian geometry, is a tool to study finitely presented groups. Dehn functions can detect various properties of groups; for example, hyperbolicity of groups. They also provide upper bounds on the complexity of the word problem for finitely presented groups. In this talk, I will survey some classical results on Dehn functions. Then I will focus on the Dehn functions of subgroups of right-angled Artin groups.

Rachel Skipper
Finiteness Properties for Simple Groups

A group is said to be of type $F_n$ if it admits a classifying space with compact $n$-skeleton. We will consider the class of R\"{o}ver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type $F_{n-1}$ but not $F_n$ for each $n$. These are the first known examples for $n\geq 3$. As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups. This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky.

Michael Landry
Flows, surfaces, and the Thurston norm

Let M be a closed hyperbolic three-manifold fibering over the circle. By work of Fried, for every (open) fibered face F of the Thurston norm ball in H_2(M;R) there is a pseudo-Anosov flow on M which collates the fibrations corresponding to F. Mosher extended this picture with his Transverse Surface Theorem, which gives a characterization of integral homology classes lying in the closure of cone(F) in terms of this flow. We will discuss a strengthening of the Transverse Surface Theorem which gives a statement about all isotopy classes of incompressible surfaces representing homology classes in the closure of cone(F) in terms of the flow associated to F. We restrict to the fibered case here for simplicity, but our theorem holds in a more general setting. The main ideas in the proof concern veering triangulations, introduced by Agol. My goal is for the talk to be relatively self-contained so the content will skew toward statements of results, and explanations of many of the above words. This is joint work with Samuel Taylor.

Claire Merriman
Coding geodesic flows and various continued fractions

I will connect continued fractions with even or odd partial quotients to geodesic flows on modular surfaces. The connection between geodesics on the modular surface PSL(2,Z)\H and regular continued fractions was established by Series. We extend this construction to the even continued fractions using the Theta-group and to the odd and grotesque continued fractions using Z/3*Z/3. This is joint work with Florin Boca.

Chandrika Sadanand
You can `hear' the shape of a polygonal billiard table

Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge-itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

Friday Geom/Top Seminar:
Emily Stark
Action rigidity for free products of hyperbolic manifold groups

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.