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

Feng Ji and Shengkui Ye

Algebraic & Geometric Topology 17 (2017) 2825–2840

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 bi(2)(G) vanishes when G is the general linear group GLn(R), the special linear group SLn(R) or the group En(R) generated by elementary matrices. When R is an infinite principal ideal domain, similar results are obtained when G is the symplectic group Sp2n(R), the elementary symplectic group ESp2n(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.

$L^2$-Betti number, acylindrical hyperbolicity, matrix groups
Mathematical Subject Classification 2010
Primary: 20F65
Received: 15 August 2016
Revised: 18 February 2017
Accepted: 27 February 2017
Published: 19 September 2017
Feng Ji
Nanyang Technological University
Shengkui Ye
Department of Mathematical Sciences
Xi’an Jiaotong-Liverpool University