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Coarse fundamental groups and box spaces

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  30 January 2019

Thiebout Delabie  and
Ana Khukhro
Thiebout Delabie
Affiliation:
Institut de Mathématiques, Université de Neuchâtel, Switzerland (thiebout.delabie@unine.ch)
Ana Khukhro
Affiliation:
Institut de Mathématiques, Université de Neuchâtel, Switzerland (anastasia.khukhro@unine.ch)
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Abstract

We use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence, we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations (Ni) and (Mi) of a free group F such that Mi > Ni for all i with [Mi:Ni] uniformly bounded, but with $\squ _{(N_i)}F$ not coarsely equivalent to $\squ _{(M_i)}F$. Finally, we give some applications of the main theorem for rank gradient and the first ℓ2 Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.

Keywords

Coarse geometrycoarse fundamental groupbox spaces

MSC classification

Primary: 20F69: Asymptotic properties of groups
Secondary: 20F34: Fundamental groups and their automorphisms 20E26: Residual properties and generalizations; residually finite groups 20F65: Geometric group theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1139 - 1154
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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