A coarse embedding theorem for homological filling functions

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.M S C ( 2 0 2 0 ) 20E07, 20J05, 57M07 (primary) INTRODUCTIONA coarse embedding of one metric space into another generalizes the notion of a quasi-isometry.It allows for the metric to be distorted by any functions that tend to infinity, rather than just linear functions allowed in quasi-isometries.Definition 1.1. We say that a map 𝑓 ∶ 𝑋 → 𝑌 is a coarse embedding if there are functions 𝜌 − , 𝜌 + ∶ ℝ → ℝ such that lim 𝑥→∞ 𝜌 − (𝑥) = lim 𝑥→∞ 𝜌 + (𝑥) = ∞ and where the following inequality holds:𝜌 − (𝑑(𝑥, 𝑦)) ⩽ 𝑑(𝑓(𝑥), 𝑓(𝑦)) ⩽ 𝜌 + (𝑑(𝑥, 𝑦)).In geometric group theory, a coarse embedding is a geometric version of subgroup containment. Indeed, if 𝐻 is a finitely generated subgroup of 𝐺, then the inclusion map is a coarse embed-