Groups with no coarse embeddings into hyperbolic groups

TL;DR

This paper introduces an obstruction called 'admitting exponentially many fat bigons' that prevents certain groups from coarsely embedding into hyperbolic groups, expanding understanding of geometric group theory.

Contribution

It defines a new coarse geometric obstruction and demonstrates its effectiveness in distinguishing groups that cannot embed into hyperbolic groups.

Findings

Groups with exponential growth and linear divergence cannot embed into hyperbolic groups.

The obstruction is preserved under coarse embeddings between bounded degree graphs.

Examples include certain lacunary hyperbolic and small cancellation groups.

Abstract

We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is "admitting exponentially many fat bigons", and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the former cannot be coarsely embedded into the latter. Other examples include certain lacunary hyperbolic and certain small cancellation groups.