Algebraic & Geometric Topology

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

Abstract

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

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

Dates
Revised: 18 February 2017
Accepted: 27 February 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841484

Digital Object Identifier
doi:10.2140/agt.2017.17.2825

Mathematical Reviews number (MathSciNet)
MR3704244

Zentralblatt MATH identifier
06791385

Subjects

Citation

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. https://projecteuclid.org/euclid.agt/1510841484

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