A coarse embedding theorem for homological filling functions

TL;DR

This paper establishes a relationship between coarse embeddings and homological Dehn functions, providing new characterizations of groups based on their embedding properties and Dehn functions.

Contribution

It proves that coarse embeddings induce inequalities in homological Dehn functions and applies this to characterize groups with specific embedding and Dehn function properties.

Findings

Coarse embeddings induce inequalities in homological Dehn functions.

Characterization of groups admitting coarse embeddings into hyperbolic groups of dimension 2.

Identification of subgroups with quadratic Dehn functions and their embedding properties.

Abstract

We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension $2$, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension $2$, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.