  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.
