On acylindrical hyperbolicity of groups with positive first ${\ell }^2$-Betti number

Abstract

We prove that every finitely presented group with positive first ${\ell }^2$-Betti number that virtually surjects onto $\mathbb{Z}$ is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented residually finite groups with positive first ${\ell }^2$-Betti number as well as groups of deficiency at least $2$.

The published version of this paper used a result from arXiv:1310.6289 whose proof contained a gap. The gap and necessary changes to the published version of this paper are discussed in arXiv:1711.09486. This version incorporates these changes

Cited by in corpus (4)

• Hyperbolic structures on groups
• Acylindrical group actions on quasi-trees
• The first ${\ell }^2$-Betti number and groups acting on trees
• Acylindrical hyperbolicity of cubical small-cancellation groups