Second bounded cohomology of groups acting on 1-manifolds and applications to spectrum problems
Open access
Date
2023-09-01Type
- Journal Article
Abstract
We prove a general criterion for the vanishing of second bounded cohomology (with trivial real coefficients) for groups that admit an action satisfying certain mild hypotheses. This leads to new computations of the second bounded cohomology for a large class of groups of homeomorphisms of 1-manifolds, and a plethora of applications. First, we demonstrate that the finitely presented and nonamenable group G0 constructed by the second author with Justin Moore satisfies that every subgroup has vanishing second bounded cohomology. This provides the first solution to a homological version of the von Neumann–Day Problem, posed by Calegari. Next, we develop a technical refinement of our criterion to demonstrate the existence of finitely generated non-indicable (even simple) left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Navas. Then we provide the first examples of finitely presented groups whose spectrum of stable commutator length contains algebraic irrationals, answering a question of Calegari. Finally, we provide the first examples of manifolds whose simplicial volumes are algebraic and irrational, as further evidence towards a conjecture of Heuer and Löh. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000620438Publication status
publishedExternal links
Journal / series
Advances in MathematicsVolume
Pages / Article No.
Publisher
ElsevierSubject
Bounded cohomology; Left orderable groups; Groups acting on the circle; Stable commutator length; Simplicial volumeOrganisational unit
08802 - Iozzi, Alessandra (Tit.-Prof.)
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Is new version of: http://hdl.handle.net/20.500.11850/527038
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