Volume distortion in groups

TL;DR

This paper introduces the concept of volume distortion in groups, showing it as a quasi-isometry invariant and exploring its properties, bounds, and specific examples including a proof of Gersten's conjecture.

Contribution

It defines volume distortion functions for group pairs, proves their invariance under quasi-isometry, and computes these functions in key examples, including a new proof of Gersten's conjecture.

Findings

Volume distortion functions are quasi-isometry invariants.

Bounds relate volume distortion to Dehn functions.

Explicit calculation of volume distortion for $ extbf{Z}^k$ in semi-direct products.

Abstract

Given a space $Y$ in $X$, a cycle in $Y$ may be filled with a chain in two ways: either by restricting the chain to $Y$ or by allowing it to be anywhere in $X$. When the pair $(G,H)$ acts on $(X, Y)$, we define the $k$-volume distortion function of $H$ in $G$ to measure the large-scale difference between the volumes of such fillings. We show that these functions are quasi-isometry invariants, and thus independent of the choice of spaces, and provide several bounds in terms of other group properties, such as Dehn functions. We also compute the volume distortion in a number of examples, including characterizing the $k$-volume distortion of $\Z^k$ in $\Z^k \rtimes_M \Z$, where $M$ is a diagonalizable matrix. We use this to prove a conjecture of Gersten.