Let be an infinite commutative ring with identity and an integer. We prove that for each integer , the –Betti number vanishes when is the general linear group , the special linear group or the group generated by elementary matrices. When is an infinite principal ideal domain, similar results are obtained when is the symplectic group , the elementary symplectic group , the split orthogonal group or the elementary orthogonal group . Furthermore, we prove that is not acylindrically hyperbolic if . We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of –rigid rings.
"Vanishing of $L^2$–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings." Algebr. Geom. Topol. 17 (5) 2825 - 2840, 2017. https://doi.org/10.2140/agt.2017.17.2825