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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

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

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

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1435-5337
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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

## Abstract

The space of configurations of n ordered points in the plane serves as a classifying space for the pure braid group $P{B}_{n}$. 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 $\mathcal{𝒞}F$ was considered of configurations of points on the real line allowing for splitting and merging, and a sketch of a proof was given that $\mathcal{𝒞}F$ 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 $CAT\left(0\right)$ cube complex X on which F acts freely, and construct an explicit homotopy equivalence between $X/F$ and $\mathcal{𝒞}F$, proving that $\mathcal{𝒞}F$ is indeed a $K\left(F,1\right)$.

MSC 2010: 20F65; 57M07; 55R80

## References

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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,

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