# Departmental PhD Thesis Exam

Event Information Pseudo-Anosov homeomorphisms constructed using positive Dehn twists
10:00 on Monday March 30, 2020
11:00 on Monday March 30, 2020
See Abstract
Yvon Verberne
University of Toronto

Location: BA2139

We introduce a construction of pseudo-Anosov homeomorphisms on $n$-times punctured spheres and surfaces with higher genus using only sufficiently many positive half-twists. These constructions can produce explicit examples of pseudo-Anosov maps with various number-theoretic properties associated to the stretch factors, including examples where the trace field is not totally real and the Galois conjugates of the stretch factor are on the unit circle.
We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism $f$, a sequence of points $w_k$ and a sequence of radii $r_k$ so that the ball $B(w_k, r_k)$ is disjoint from a quasi-axis $a$ of $f$, but for any projection map from the mapping class group to $a$, the diameter of the image of $B(w_k, r_k)$ grows like $\log(r_k)$.