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Braided Thompson groups with and without quasimorphisms

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

Algebraic & Geometric Topology 24 (2024) 1601–1622

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 bV has an infinite-dimensional space of quasimorphisms and thus infinite-dimensional second bounded cohomology. This implies that, despite being perfect, bV is not uniformly perfect, in contrast to Thompson’s group V . We also prove that relatives of bV like the ribbon braided Thompson group rV and the pure braided Thompson group bF similarly have an infinite-dimensional space of quasimorphisms. Our second main result is that, in stark contrast, the close relative of bV denoted by bV ^, which was introduced concurrently by Brin, has trivial second bounded cohomology. This makes bV ^ the first example of a left-orderable group of type F that is not locally indicable and has trivial second bounded cohomology. This also makes bV ^ 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.

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
Received: 18 April 2022
Accepted: 25 August 2022
Published: 28 June 2024
Francesco Fournier-Facio
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom
Yash Lodha
Department of Mathematics
University of Hawai‘i at Mānoa
Honolulu, HI
United States
Matthew C B Zaremsky
Department of Mathematics and Statistics
University at Albany (SUNY)
Albany, NY
United States

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