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 $n\ge 2$ be an integer. We prove that for each integer $i=0,1,\cdots ,n-2,$ the $L^2$-Betti number $b_i^{(2)}(G)=0,$  when $G={\mathrm{GL}}_n(R)$ the general linear group, ${\mathrm{SL}}_n(R)$ the special linear group, the group generated by elementary matrices. When $R$ is an infinite principal ideal domain, similar results are obtained for ${\mathrm{Sp}}_{2n}(R)$ the symplectic group, ${\mathrm{ESp}}_{2n}(R)$ the elementary symplectic group, $\mathrm{O}(n,n)(R)$ the split orthogonal group or $\mathrm{EO}(n,n)(R)$ the elementary orthogonal group. 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.

17 pages, to appear in Algebraic & Geometric Topology

References in corpus (1)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces