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Let $R$ be an infinite commutative ring with identity and $n\geq 2$ be aninteger. We prove that for each integer $i=0,1,\cdots ,n-2,$ the $L^{2}$-Bettinumber $b_{i}^{(2)}(G)=0,$ $\ $when $G=\mathrm{GL}_{n}(R)$ the general lineargroup, $\mathrm{SL}_{n}(R)$ the special linear group, $% E_{n}(R)$ the groupgenerated by elementary matrices. When $R$ is an infinite principal idealdomain, similar results are obtained for $\mathrm{Sp}_{2n}(R)$ the symplecticgroup, $\mathrm{ESp}_{2n}(R)$ the elementary symplectic group,$\mathrm{O}(n,n)(R)$ the split orthogonal group or $\mathrm{EO}(n,n)(R)$ theelementary orthogonal group. Furthermore, we prove that $G$ is notacylindrically hyperbolic if $n\geq 4$. We also prove similar results for aclass of noncommutative rings. The proofs are based on a notion of $n$-rigidrings.

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