Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 17, Number 5 (2017), 2825-2840.
Vanishing of $L^2$–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings
Let be an infinite commutative ring with identity and an integer. We prove that for each integer , the –Betti number vanishes when is the general linear group , the special linear group or the group generated by elementary matrices. When is an infinite principal ideal domain, similar results are obtained when is the symplectic group , the elementary symplectic group , the split orthogonal group or the elementary orthogonal group . Furthermore, we prove that is not acylindrically hyperbolic if . We also prove similar results for a class of noncommutative rings. The proofs are based on a notion of –rigid rings.
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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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
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