Vanishing of L²–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings

Let R be an infinite commutative ring with identity and n ≥ 2 an integer. We prove that for each integer i = 0, 1,…, n − 2, the L²–Betti number b_i ⁽²⁾(G) vanishes when G is the general linear group GL_n(R), the special linear group SL_n(R) or the group E_n(R) generated by elementary matrices. When R is an infinite principal ideal domain, similar results are obtained when G is the symplectic group Sp_2n(R), the elementary symplectic group ESp_2n(R), the split orthogonal group O(n, n)(R) or the elementary orthogonal group EO(n, n)(R). Furthermore, we prove that G is not acylindrically hyperbolic if n ≥ 4. We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of n–rigid rings.