Braided Thompson groups with and without quasimorphisms

Fournier-Facio, Francesco¹ ; Lodha, Yash² ; Zaremsky, Matthew C B³

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, United Kingdom
² Department of Mathematics, University of Hawai‘i at Mānoa, Honolulu, HI, United States
³ Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 3, pp. 1601-1622

Résumé

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.

Détail

DOI : 10.2140/agt.2024.24.1601

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