Volume 19, issue 3 (2019)

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Classifying spaces from Ore categories with Garside families

Stefan Witzel

Algebraic & Geometric Topology 19 (2019) 1477–1524
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.

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