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

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).

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