Projections in the curve complex arising from covering maps

Robert Tang

Vol. 291 (2017), No. 1, 213–239

DOI: 10.2140/pjm.2017.291.213

Abstract

Let $P:\Sigma \to S$ be a finite degree covering map between surfaces. Rafi and Schleimer showed that there is an induced quasi-isometric embedding $\Pi :\mathcal{C}\left(S\right)\to \mathcal{C}\left(\Sigma \right)$ between the associated curve complexes. We define an operation on curves in $\mathcal{C}\left(\Sigma \right)$ using minimal intersection number conditions and prove that it approximates a nearest point projection to $\Pi \left(\mathcal{C}\left(S\right)\right)$. We also approximate hulls of finite sets of vertices in the curve complex, together with their corresponding nearest point projections, using intersection numbers.

Keywords

curve complex, covering map, nearest point projection, hull

Mathematical Subject Classification 2010

Primary: 20F65, 57M50

Milestones

Received: 6 November 2014

Revised: 11 June 2016

Accepted: 28 December 2016

Published: 20 August 2017