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Forum Mathematicum

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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 29, Issue 3


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

Lucas Sabalka / Matthew C. B. Zaremsky
Published Online: 2016-08-19 | DOI: https://doi.org/10.1515/forum-2015-0148


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).

Keywords: Thompson’s group; classifying space; configuration space

MSC 2010: 20F65; 57M07; 55R80


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About the article

Received: 2015-07-27

Published Online: 2016-08-19

Published in Print: 2017-05-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: SFB 701

Award identifier / Grant number: SFB 878

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

Citation Information: Forum Mathematicum, Volume 29, Issue 3, Pages 681–691, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0148.

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