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On Belk’s classifying space for Thompson’s group F

  • Lucas Sabalka and Matthew C. B. Zaremsky EMAIL logo
Published/Copyright: August 19, 2016

Abstract

The space of configurations of n ordered points in the plane serves as a classifying space for the pure braid group PBn. Elements of Thompson’s group F admit a model similar to braids, except instead of braiding the strands split and merge. In Belk’s thesis, a space 𝒞F was considered of configurations of points on the real line allowing for splitting and merging, and a sketch of a proof was given that 𝒞F is a classifying space for F. The idea there was to build the universal cover and construct an explicit contraction to a point. However, this was never written up rigorously. Here we start with an established CAT(0) cube complex X on which F acts freely, and construct an explicit homotopy equivalence between X/F and 𝒞F, proving that 𝒞F is indeed a K(F,1).

MSC 2010: 20F65; 57M07; 55R80

Communicated by Frederick R. Cohen


Award Identifier / Grant number: SFB 701

Award Identifier / Grant number: SFB 878

Funding statement: The second author was supported by the SFB 701 in Bielefeld and SFB 878 in Münster during the course of this work.

Acknowledgements

The first author thanks Keith Jones for the discussions that led to this project. The second author thanks Kai-Uwe Bux and Stefan Witzel for helpful discussions and suggestions. We are also grateful to Jim Belk and Ross Geoghegan for helpful conversations. Finally, we wish to thank two anonymous referees: the first for catching a critical error in a previous version of this paper, and the second for a thorough reading of the current version.

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Received: 2015-7-27
Published Online: 2016-8-19
Published in Print: 2017-5-1

© 2017 by De Gruyter

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