Algebraic & Geometric Topology

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

Feng Ji and Shengkui Ye

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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.

Article information

Algebr. Geom. Topol., Volume 17, Number 5 (2017), 2825-2840.

Received: 15 August 2016
Revised: 18 February 2017
Accepted: 27 February 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

$L^2$-Betti number acylindrical hyperbolicity matrix groups


Ji, Feng; Ye, Shengkui. Vanishing of $L^2$–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings. Algebr. Geom. Topol. 17 (2017), no. 5, 2825--2840. doi:10.2140/agt.2017.17.2825.

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