Projections of the sphere graph to the arc graph of a surface

Abstract

Let [Formula: see text] be a compact surface, and [Formula: see text] be the double of a handlebody. Given a homotopy class of maps from [Formula: see text] to [Formula: see text] inducing an isomorphism of fundamental groups, we describe a canonical uniformly Lipschitz retraction of the sphere graph of [Formula: see text] to the arc graph of [Formula: see text]. We also show that this retraction is a uniformly bounded distance from the nearest point projection map.