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2017 Vanishing of $L^2$–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings
Feng Ji, Shengkui Ye
Algebr. Geom. Topol. 17(5): 2825-2840 (2017). DOI: 10.2140/agt.2017.17.2825

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.

Citation

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Feng Ji. Shengkui Ye. "Vanishing of $L^2$–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings." Algebr. Geom. Topol. 17 (5) 2825 - 2840, 2017. https://doi.org/10.2140/agt.2017.17.2825

Information

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

zbMATH: 06791385
MathSciNet: MR3704244
Digital Object Identifier: 10.2140/agt.2017.17.2825

Subjects:
Primary: 20F65

Rights: Copyright © 2017 Mathematical Sciences Publishers

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