Friday, January 24th, 2020

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Rylee Lyman (Tufts University)

Title: Train tracks and pseudo-Anosov braids in automorphisms of free products.

Abstract: The Nielsen–Thurston classification of surface homeomorphisms says that every homeomorphism of a surface either has a finite power isotopic to the identity, preserves the isotopy class of some essential multi-curve, or is isotopic to a pseudo-Anosov map, the most interesting kind. Bestvina and Handel introduced a similar classification for automorphisms of free groups. Here the analogue of a pseudo-Anosov homeomorphism is a train track map for an outer automorphism which is fully irreducible, a homotopy equivalence of a graph with extra structure. The analogy really is correct: pseudo-Anosov mapping classes of once-punctured surfaces induce fully irreducible outer automorphisms preserving a nontrivial conjugacy class and vice-versa. We discuss extensions of the train track theory to automorphisms of free products. Here the analogy is to mapping classes of punctured spheres. We show that fully irreducible automorphisms of free products of finite subgroups of SO(2) may be represented as pseudo-Anosov braids on orbifolds if and only if they preserve a non-peripheral conjugacy class.

Friday, November 22nd, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Neil MacVicar (Queen's University)

Title: Bratteli Diagrams and Cantor Minimal Systems.

Abstract: A Bratteli diagram is a kind of infinite graph for which a transformation can be defined on its path space. This talk will introduce the diagrams, their associated dynamical systems, and the relationship between these systems and systems described by a minimal homeomorphism acting on a Cantor space.

Friday, November 1st, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Giulio Tiozzo (Queen's University)

Title: Entropy and drift for Gibbs measures on geometrically finite manifolds.

Abstract: The boundary of a simply connected, negatively curved manifold carries two natural types of measures: on one hand, Gibbs measures such as the Patterson-Sullivan measure and the SRB measure. On the other hand, harmonic measures arising from random walks. We prove that the absolute continuity between a harmonic measure and a Gibbs measure is equivalent to a relation between entropy, drift and critical exponent, extending the previous formulas of Guivarc’h, Ledrappier, and Blachere-Haissinsky-Mathieu. This shows that if the manifold (or more generally, a CAT(-1) space) is geometrically finite but not convex cocompact, harmonic measures are singular with respect to Gibbs measures.

Friday, September 20th, 2019

Time: 10:30 a.m Place: Jeffery Hall 319

Speaker: Ian Frankel (Queen's University)

Title: Local geometry of Teichmüller space: flat and quasiconformal.

Abstract: The Teichmüller distance between two homeomorphic Riemann surfaces X and Y is a number that quantifies the following question: Given a homeomorphism from X to Y, how non-conformal does the map have to be?

The optimal quasiconformal maps, i.e. those with smallest quasiconformal constant, are characterized by choices of special singular flat metrics on X and Y, and in fact fit into a large familes of maps, and the dynamics of SL(2,R) acting on this family have been the subject of many celebrated results in the past decade.

Now, suppose we are given X and Y but with singular flat metrics that are not related to the optimal map. We will describe how we can still estimate the Teichmüller distance from X to Y.

Tuesday, August 27th, 2019

Time: 1:30 p.m Place: Jeffery Hall 319

Speaker: Catherine Pfaff (Queen's University)

Title: Typical Trees: An Out(F_r) Excursion.

Abstract: Random walks are not new to geometric group theory (see, for example, work of Furstenberg, Kaimonovich, Masur). However, following independent proofs by Maher and Rivin that pseudo-Anosovs are generic within mapping class groups, and then new techniques developed by Maher-Tiozzo, Sisto, and others, the field has seen in the past decade a veritable explosion of results. In a 2 paper series, we answer with fine detail a question posed by Handel-Mosher asking about invariants of generic outer automorphisms of free groups and then a question posed by Bestvina as to properties of R-trees of full measure in the boundary of Culler-Vogtmann outer space. This is joint work with Ilya Kapovich, Joseph Maher, and Samuel J. Taylor.