### Volume distortion in groupsVolume distortion in groups

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 Author Bennett, Hanna Source arXiv.org Content type Text File Format PDF Date of Submission 2010-02-04 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Group Theory ♦ Mathematics - Geometric Topology ♦ 20F65 ♦ math 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. Description Reference: Algebr. Geom. Topol. 11 (2011) 655-690 Educational Use Research Learning Resource Type Article Page Count 27