#### Volume 11, issue 2 (2011)

 Recent Issues
Author Index
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
Volume distortion in groups

### Hanna Bennett

Algebraic & Geometric Topology 11 (2011) 655–690
##### 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 $\left(G,H\right)$ acts on $\left(X,Y\right)$, 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 ${ℤ}^{k}{⋊}_{M}ℤ$, where $M$ is a diagonalizable matrix. We use this to prove a conjecture of Gersten.

##### Keywords
geometric group theory, volume distortion, subgroup distortion, Dehn function
##### Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20F67, 57M07