Hyperbolic spaces not quasi-isometric to curve complexes

TL;DR

This paper identifies conditions under which certain hyperbolic spaces cannot be quasi-isometric to curve complexes, impacting the understanding of geometric structures associated with surfaces and free groups.

Contribution

It establishes a criterion that rules out quasi-isometry between hyperbolic spaces and various complexes related to surfaces and free groups.

Findings

Hyperbolic complexes like arc, disk, and pants complexes are not quasi-isometric to curve complexes under certain conditions.

The result applies to free splitting complexes of free groups.

Provides a unifying condition affecting multiple hyperbolic complexes.

Abstract

We identify a condition that prevents a hyperbolic space from being quasi-isometric to the curve complex of any non-sporadic surface. Our result applies to several hyperbolic complexes, including arc complexes, disk complexes, non-separating curve complexes, (hyperbolic) pants complexes, and to free splitting complexes of free groups.