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

TL;DR

This paper proves that finitely presented groups with positive first $ ext{l}^2$-Betti number that virtually surject onto $ ext{Z}$ are acylindrically hyperbolic, extending to residually finite groups and those with deficiency at least 2.

Contribution

It establishes a broad criterion linking positive first $ ext{l}^2$-Betti number and virtual surjection onto $ ext{Z}$ to acylindrical hyperbolicity in finitely presented groups.

Findings

Finitely presented groups with positive first $ ext{l}^2$-Betti number and a virtual surjection onto $ ext{Z}$ are acylindrically hyperbolic.

Residually finite groups with positive first $ ext{l}^2$-Betti number are acylindrically hyperbolic.

Groups with deficiency at least 2 are acylindrically hyperbolic.

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 $2$.