Braided Thompson groups with and without quasimorphisms

Fournier-Facio, Francesco; Lodha, Yash; Zaremsky, Matthew C B

doi:10.2140/agt.2024.24.1601

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

Braided Thompson groups with and without quasimorphisms

Francesco Fournier-Facio, Yash Lodha and Matthew C B Zaremsky

Algebraic & Geometric Topology 24 (2024) 1601–1622

DOI: 10.2140/agt.2024.24.1601

Abstract

We study quasimorphisms and bounded cohomology of a variety of braided versions of Thompson groups. Our first main result is that the Brin–Dehornoy braided Thompson group $b V$ has an infinite-dimensional space of quasimorphisms and thus infinite-dimensional second bounded cohomology. This implies that, despite being perfect, $b V$ is not uniformly perfect, in contrast to Thompson’s group $V$ . We also prove that relatives of $b V$ like the ribbon braided Thompson group $r V$ and the pure braided Thompson group $b F$ similarly have an infinite-dimensional space of quasimorphisms. Our second main result is that, in stark contrast, the close relative of $b V$ denoted by $\hat{b V}$ , which was introduced concurrently by Brin, has trivial second bounded cohomology. This makes $\hat{b V}$ the first example of a left-orderable group of type $F_{\infty}$ that is not locally indicable and has trivial second bounded cohomology. This also makes $\hat{b V}$ an interesting example of a subgroup of the mapping class group of the plane minus a Cantor set that is nonamenable but has trivial second bounded cohomology, behavior that cannot happen for finite-type mapping class groups.

Keywords

braid group, Thompson group, quasimorphism, bounded cohomology, uniformly perfect, big mapping class group, left-orderable group

Mathematical Subject Classification

Primary: 20F65, 20J05

Secondary: 20F36, 57K20

References

Publication

Received: 18 April 2022

Accepted: 25 August 2022

Published: 28 June 2024

Authors

Francesco Fournier-Facio
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge United Kingdom
https://www.fffmaths.com
Yash Lodha
Department of Mathematics University of Hawai‘i at Mānoa Honolulu, HI United States
https://yl7639.wixsite.com/website
Matthew C B Zaremsky
Department of Mathematics and Statistics University at Albany (SUNY) Albany, NY United States
https://www.albany.edu/~mz498674/

Open Access made possible by participating institutions via Subscribe to Open.

Volume 24, issue 3 (2024)