Volume distortion in groups

Bennett, Hanna¹

¹Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church St, Ann Arbor MI 48109, USA

Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 655-690

Résumé

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

Détail

DOI : 10.2140/agt.2011.11.655

Keywords: geometric group theory, volume distortion, subgroup distortion, Dehn function

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