Event
Yvon Verberne, University of Toronto
Wednesday, April 11, 2018 15:00to16:00
Burnside Hall
Room 920, 805 rue Sherbrooke Ouest, Montreal, QC, H3A 0B9, CA
Title: Strong contractibility in the mapping class group.
The mapping class group of a compact surface $S$, denoted $MCG(S)$, is the group of isotopy classes of elements of $Homeo^{+}(S, partial S)$, where isotopies are required to fix the boundary pointwise. The complex of curves, $C(S)$, is a simplicial complex where the vertices are free isotopy classes of essential simple closed curves in $S$ and there is an edge connecting two vertices if the isotopy classes have geometric intersection number zero. Choosing a generating set defines a word metric on the mapping class group. By using the combinatorial properties of $C(S)$ along with the action of $MCG(S)$ on $C(S)$ we are able to study the geometry of the mapping class group. Previously, Masur and Minsky constructed quasi-geodesics in $MCG(S)$, but not much is known about geodesics. In joint work with Kasra Rafi, we have found some explicit examples of geodesics, denoted $gamma$, in $MCG(S_{0,5})$, where $S_{0,5}$ is the five-times punctured sphere. We were able to find these geodesics by having constructed an appropriate generating set and having found a homomorphism from $MCG(S_{0,5})$ to $mathbb{Z}$. In addition, we have constructed a pseudo-Anosov map whose axis is not strongly contracting. A geodesic is strongly contracting if its nearest point projection takes disjoint balls from the geodesic to sets of bounded diameter, where the bound should be independent of the ball. Strongly contracting geodesics show up in the study of growth tightness, for example, the strongly contracting property was used to prove growth tightness for actions on non relatively hyperbolic spaces in work by G. Arzhantseva, C. Cashen, J. Tao, which is why we pursued this result.