Word hyperbolic extensions of surface groups

TL;DR

This paper characterizes when extensions of surface groups are word hyperbolic by linking hyperbolicity to the quasi-isometric embedding of the orbit map on the complex of curves, connecting it to convex cocompactness.

Contribution

It establishes a precise criterion for hyperbolicity of surface group extensions via the orbit map on the complex of curves and convex cocompactness.

Findings

Word hyperbolic extensions correspond to quasi-isometric orbit maps.

Hyperbolicity is equivalent to convex cocompactness in the quotient group.

Provides a geometric criterion for hyperbolicity of surface group extensions.

Abstract

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex cocompact in the sense of Farb and Mosher.