Classifying spaces from Ore categories with Garside families

Witzel, Stefan

doi:10.2140/agt.2019.19.1477

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Classifying spaces from Ore categories with Garside families

Stefan Witzel

Algebraic & Geometric Topology 19 (2019) 1477–1524

DOI: 10.2140/agt.2019.19.1477

Abstract

We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady’s classifying space for braid groups and the Stein–Farley complexes used for various relatives of Thompson’s groups. It recovers the fact that Garside groups have finite classifying spaces.

We describe the categories and Garside structures underlying certain Thompson groups. The indirect product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our methods we introduce the group braided $T$ and show that it is of type $F_{\infty}$ .

Keywords

Thompson groups, braid groups, finiteness properties, classifying spaces, Ore categories, Garside structures

Mathematical Subject Classification 2010

Primary: 57M07

Secondary: 20F36, 20F65

References

Publication

Received: 27 April 2018

Revised: 1 November 2018

Accepted: 1 November 2018

Published: 21 May 2019

Authors

Stefan Witzel
Faculty of Mathematics Bielefeld University Bielefeld Germany

Volume 19, issue 3 (2019)