Vanishing of $L^2$–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings

Abstract

Let $R$ be an infinite commutative ring with identity and $n\ge 2$ an integer. We prove that for each integer $i=0,1,\dots ,n-2$, the $L^2$–Betti number $b_i^{\left(2\right)}\left(G\right)$ vanishes when $G$ is the general linear group ${GL}_n\left(R\right)$, the special linear group ${SL}_n\left(R\right)$ or the group $E_n\left(R\right)$ generated by elementary matrices. When $R$ is an infinite principal ideal domain, similar results are obtained when $G$ is the symplectic group ${Sp}_{2n}\left(R\right)$, the elementary symplectic group ${ESp}_{2n}\left(R\right)$, the split orthogonal group $O\left(n,n\right)\left(R\right)$ or the elementary orthogonal group $EO\left(n,n\right)\left(R\right)$. Furthermore, we prove that $G$ is not acylindrically hyperbolic if $n\ge 4$. We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of $n$–rigid rings.