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