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

Ji, Feng; Ye, Shengkui

doi:10.2140/agt.2017.17.2825

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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

DOI: 10.2140/agt.2017.17.2825

Abstract

Let $R$ be an infinite commutative ring with identity and $n \geq 2$ an integer. We prove that for each integer $i = 0, 1, \dots, n - 2$ , the $L^{2}$ –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 ${Sp}_{2 n} (R)$ , the elementary symplectic group ${ESp}_{2 n} (R)$ , 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 $n \geq 4$ . We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of $n$ –rigid rings.

Keywords

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

Mathematical Subject Classification 2010

Primary: 20F65

References

Publication

Received: 15 August 2016

Revised: 18 February 2017

Accepted: 27 February 2017

Published: 19 September 2017

Authors

Feng Ji
Infinitus Nanyang Technological University Singapore

Shengkui Ye
Department of Mathematical Sciences Xi’an Jiaotong-Liverpool University Jiangsu China
https://yeshengkui.wordpress.com/

Volume 17, issue 5 (2017)