Volume 11, issue 2 (2011)

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Volume distortion in groups

Hanna Bennett

Algebraic & Geometric Topology 11 (2011) 655–690

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 k in k M, where M is a diagonalizable matrix. We use this to prove a conjecture of Gersten.

geometric group theory, volume distortion, subgroup distortion, Dehn function
Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20F67, 57M07
Received: 12 February 2010
Revised: 21 December 2010
Accepted: 21 December 2010
Published: 1 March 2011
Hanna Bennett
Department of Mathematics
University of Michigan
2074 East Hall
530 Church St
Ann Arbor MI 48109