Volume distortion in groups

Bennett, Hanna

doi:10.2140/agt.2011.11.655

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Volume distortion in groups

Hanna Bennett

Algebraic & Geometric Topology 11 (2011) 655–690

DOI: 10.2140/agt.2011.11.655

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 $ℤ^{k} ⋊_{M} ℤ$ , 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

References

Publication

Received: 12 February 2010

Revised: 21 December 2010

Accepted: 21 December 2010

Published: 1 March 2011

Authors

Hanna Bennett
Department of Mathematics University of Michigan 2074 East Hall 530 Church St Ann Arbor MI 48109 USA
http://www.math.lsa.umich.edu/~hbennett/

Volume 11, issue 2 (2011)