Volume distortion in groups

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 ${\mathbb{Z}}^k$ in ${\mathbb{Z}}^k{⋊}_M\mathbb{Z}$, where $M$ is a diagonalizable matrix. We use this to prove a conjecture of Gersten.