Hyperbolicity and bounded-valued cohomology

TL;DR

This paper extends a theorem on bounded cohomology of groups, providing new criteria for hyperbolicity, applications to subgroup analysis, and calculations for specific classes of groups, advancing understanding in geometric group theory.

Contribution

It generalizes Gersten's theorem on $ ext{ell}^ olinebreak^{ ext{infty}}$-cohomology, introduces hyperbolicity criteria for certain groups, and offers new applications and calculations in bounded cohomology.

Findings

Generalized Gersten's theorem on $ ext{ell}^ olinebreak^{ ext{infty}}$-cohomology

Established hyperbolicity criteria for groups of type $FP_2(Q)$

Provided $ ext{ell}^ olinebreak^{ ext{infty}}$-cohomology calculations for specific classes

Abstract

We generalise a theorem of Gersten on surjectivity of the restriction map in $\ell^{\infty}$-cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and $\ell^{\infty}$-cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type $FP_2(\mathbb Q)$ and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Mart\'{i}nez-Pedroza.