Finitely generated simple left orderable groups with vanishing second bounded cohomology

TL;DR

This paper demonstrates that certain finitely generated simple left orderable groups have vanishing second bounded cohomology, providing new examples that answer a longstanding question in the field.

Contribution

It introduces the first known finitely generated non-indicable left orderable groups with vanishing second bounded cohomology, expanding understanding of their algebraic and cohomological properties.

Findings

First examples of finitely generated non-indicable left orderable groups with vanishing second bounded cohomology

Proves vanishing of second bounded cohomology with trivial real and integral coefficients

Answers a question posed in the 2018 ICM proceedings

Abstract

We prove that the finitely generated simple left orderable groups constructed by the second author with Hyde have vanishing second bounded cohomology, both with trivial real and trivial integral coefficients. As a consequence, these are the first examples of finitely generated non-indicable left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Andr\'es Navas.