Stable commutator length on big mapping class groups

Field, Elizabeth H.; Patel, Priyam; Rasmussen, Alexander J.

doi:10.48550/arxiv.2108.02123

Cited by 1 publication

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“…Which elements of bV have non-zero stable commutator length? A characterisation of this phenomenon in (finite-type) mapping class groups was given in [9], and see [27] for some related results for big mapping class groups. Theorem 1.2 has interesting consequences for subgroups of big mapping class groups.…”

Section: Introductionmentioning

confidence: 99%

Braided Thompson groups with and without quasimorphisms

Fournier-Facio¹,

Lodha²,

Zaremsky³

2022

Preprint

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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 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 non-amenable but has trivial second bounded cohomology, behaviour that cannot happen for finite-type mapping class groups.

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Section: Introductionmentioning

confidence: 99%