Max Dehn Seminar

on Geometry, Topology, Dynamics, and Groups

Zoom information: Meeting ID: 980 1095 2691 Passcode: 060530

Date Speaker Title click for abstract (if available)
September 2 Elizabeth Field
University of Utah

Abstract: When $1\to H\to G\to Q\to 1$ is a short exact sequence of three word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from $H$ to $G$ extends continuously to a map between the Gromov boundaries of $H$ and $G$. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point $z$ in the Gromov boundary of $Q$ an ending lamination'' on $H$ which consists of pairs of distinct points in the boundary of $H$. We prove that for each such $z$, the quotient of the Gromov boundary of $H$ by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where $H$ is a free group and $Q$ is a convex cocompact purely atoroidal subgroup of $\mathrm{Out}(F_N)$, one can identify the resultant quotient space with a certain $\mathbb{R}$-tree in the boundary of Culler-Vogtmann's Outer space.

September 9 George Domat
University of Utah

Pure mapping class groups of finite type surfaces are known to have trivial abelianizations (perfect) once the surface has genus at least 3 due to a classic result of Powell. Aramayona-Patel-Vlamis showed that this is not always true for pure mapping class groups of infinite type surfaces. We show that this is in fact never the case for any infinite type surface. Furthermore, we show that the abelianization of the closure of the compactly supported mapping classes contains a direct summand isomorphic to an uncountable direct sum of $\mathbb{Q}$'s. To find nontrivial elements in the abelianization we use the projection complex machinery of Bestvina-Bromberg-Fujiwara to build quasimorphisms that "see" certain infinite products of Dehn Twists.

September 16 Alex Rasmussen
University of Utah

In this talk I will discuss the classification of the cobounded hyperbolic actions of solvable Baumslag-Solitar groups and Lamplighter groups. These groups have actions on trees which have convenient descriptions using ring theory. I will then discuss current work on generalizing this ring theoretic machinery, and possible applications to certain abelian-by-cyclic groups. This is joint work with Carolyn Abbott and Sahana Balasubramanya.

September 23 Tyrone Ghaswala
Université du Québec à Montréal

In the world of finite-type surfaces, one of the key tools to studying the mapping class group is to study its action on the curve graph. The curve graph is a combinatorial object intrinsic to the surface, and its appeal lies in the fact that it is infinite-diameter and δ-hyperbolic. For infinite-type surfaces, the curve graph disappointingly has diameter 2. However, all hope is not lost! In this talk I will introduce the omnipresent arc graph and we will see that for a large collection of infinite-type surfaces, the graph is infinite-diameter and δ-hyperbolic. The talk will feature a new characterization of infinite-type surfaces, which provided the impetus for this project.
This is joint work with Federica Fanoni and Alan McLeay.

September 30 Nicholas Miller
UC Berkeley

It is a consequence of the Margulis dichotomy that when an arithmetic hyperbolic manifold contains one totally geodesic hypersurface, it contains infinitely many. Both Reid and McMullen have asked conversely whether the existence of infinitely many geodesic hypersurfaces implies arithmeticity of the corresponding hyperbolic manifold. In this talk, I will discuss recent results answering this question in the affirmative. In particular, I will describe how this follows from a general superrigidity style theorem for certain natural representations of fundamental groups of hyperbolic manifolds. This is joint work with Bader, Fisher, and Stover.

University of Hawaii at Manoa

Mapping class groups of surfaces of genus at least 3 are perfect, but their finite-index subgroups need not be - they can have non-trivial abelianizations. A well-known conjecture of Ivanov states that a finite-index subgroup of a mapping class group in genus at least 3 has finite abelianization. We will discuss a proof of this conjecture, which goes through an equivalent representation-theoretic form of the conjecture due to Putman and Wieland.

October 14 Filippo Mazzoli
University of Virginia

In a recent work Bridgeman, Brock, and Bromberg characterized the infimum of the renormalized volume of a convex cocompact hyperbolic 3-manifold with incompressible boundary, as we deform its structure by quasi-isometries. In this talk I will describe a series of similarities between the renormalized volume and another notion of volume for such class of manifolds, namely the dual volume of the convex core. In particular, we will see how these analogies and the properties of Gaussian curvature surfaces allow us to obtain a similar characterization for the dual volume function.

October 21 Mark Pengitore
The Ohio State University

In this talk, we will relate homological filling functions and the existence of coarse embeddings of finitely presented groups. In particular, we will demonstrate that a coarse embedding of a finitely presented group into a group of geometric dimension 2 induces an inequality on 2 dimensional homological Dehn functions. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. If there is enough time, we will talk about various higher dimensional generalizations of our main result.

October 28 Justin Lanier
University of Chicago

There are many natural homomorphisms from mapping class groups to other groups. Are there also some surprising ones? Sometimes, but sometimes not. In this vein, we will discuss a new short proof of a theorem of Aramayona–Souto that constrains homomorphisms between mapping class groups of closed surfaces. We will also discuss new results on constraining homomorphisms from mapping class groups to homeomorphism groups of spheres. The proofs proceed by analyzing finite subgroups. This is joint work with Lei Chen.

November 4 George Domat
University of Utah

Pure mapping class groups of finite type surfaces are known to have trivial abelianizations (perfect) once the surface has genus at least 3 due to a classic result of Powell. Aramayona-Patel-Vlamis showed that this is not always true for pure mapping class groups of infinite type surfaces. We show that this is in fact never the case for any infinite type surface. Furthermore, we show that the abelianization of the closure of the compactly supported mapping classes contains a direct summand isomorphic to an uncountable direct sum of $Q$s. To find nontrivial elements in the abelianization we use the projection complex machinery of Bestvina-Bromberg-Fujiwara to build quasimorphisms that " see" certain infinite products of Dehn Twists.

November 11 Marissa Kawehi Loving
Georgia Tech

I will discuss volumes of mapping tori associated to irreducible end periodic homeomorphisms of certain infinite-type surfaces, inspired by a theorem of Brock (in the finite-type setting) relating the volume of a mapping torus to the translation distance of its monodromy on the pants graph. This talk represents joint work with Elizabeth Field, Heejoung Kim, and Chris Leininger.

November 18 Harrison Bray
George Mason

I will discuss joint work with Constantine, building on joint work with Adeboye and Constantine, on a volume-entropy rigidity result for finite volume strictly convex projective manifolds in dimension at least 3. The result is a Besson-Courtois-Gallot type theorem, using the barycenter method. As an application, we get a uniform lower bound on the Hilbert volume of a finite volume strictly convex projective manifold of dimension at least 3.

February 3 Francisco Arana-Herrera
Stanford University

Much is known about the dynamics of the mapping class group on different spaces: Teichmüller space, the space of singular measured foliations, the space of geodesic currents. However, very little is known about its effective dynamics. In this talk I will discuss work in progress that aims at clearing up this picture. Applications to counting problems on surfaces, including a partial solution to an open problem of Wright, will also be discussed. No previous knowledge of any of these topics will be assumed.

February 17 Nima Hoda
École Normale Supérieure

This talk concerns three classes of metric spaces defined by nonpositive curvature-like properties: hierarchically hyperbolic spaces, coarse Helly spaces and strongly shortcut spaces. Roughly speaking, hierarchically hyperbolic spaces are metric spaces whose geometry can be understood through a structured family of projections onto Gromov-hyperbolic spaces; coarse Helly spaces are metric spaces whose balls satisfy a coarse version of the classical 1-dimensional Helly property; and strongly shortcut spaces are metric spaces whose subspaces cannot approximate long circles with arbitrary precision.
I will discuss recent joint work with Thomas Haettel and Harry Petyt in which we prove that hierarchically hyperbolic spaces are coarsely Helly and that coarse Helly spaces of uniformly bounded geometry are strongly shortcut. This work has important consequences for hierarchically hyperbolic groups: it shows that they are semihyperbolic, have solvable conjugacy problem, are of type FP-infinity, have finitely many conjugacy classes of finite subgroups and that their finitely generated abelian subgroups are undistorted.

March 3 Ryan Dickmann
University of Utah

The mapping class groups of infinite type surfaces have become the recent focus of many researchers, but little has been said about the case of surfaces with noncompact boundary components. In this talk, we will discuss the classification of the surfaces with noncompact boundary by Brown and Messer and a new method for cutting a general surface into simpler surfaces. Using this cutting method and recent work of George Domat, we can classify the surfaces with perfect pure mapping class groups.

Archive of past talks

You may also be interested in the RTG Seminar
Max Dehn Seminar is organized by Mladen Bestvina, Ken Bromberg, Jon Chaika, Elizabeth Field,
Osama Khalil, Priyam Patel, Alex Rasmussen, Domingo Toledo, and Kevin Wortman.

This web page is maintained by Jon Chaika.