Volume 21, issue 6 (2021)

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Geodesics in the mapping class group

Kasra Rafi and Yvon Verberne

Algebraic & Geometric Topology 21 (2021) 2995–3017
Abstract

We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in the mapping class group to the curve graph does not have to be a uniform quality quasigeodesic. We also show that the quasiaxis of a pseudo-Anosov element of the mapping class group may not have the strongly contracting property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism ϕ, a sequence of points wk and a sequence of radii rk such that the ball B(wk,rk) is disjoint from a quasiaxis  aϕ of ϕ, but, for any projection from the mapping class group to aϕ, the diameter of the image of B(wk,rk) grows like log(rk).

Keywords
homeomorphisms of surfaces, mapping class group, pseudo-Anosov homeomorphisms, geodesic, contracting
Mathematical Subject Classification 2010
Primary: 20F34, 37E30, 57M07
References
Publication
Received: 4 February 2020
Revised: 26 August 2020
Accepted: 1 October 2020
Published: 22 November 2021
Authors
Kasra Rafi
Department of Mathematics
University of Toronto
Toronto, ON
Canada
Yvon Verberne
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States