## Dynamics, Geometry, & Groups - Rylee Lyman (Tufts University)

### 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.