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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.
Keywords: $L^2$-Betti number, acylindrical hyperbolicity, matrix groups
Ji, Feng 1 ; Ye, Shengkui 2
@article{10_2140_agt_2017_17_2825,
author = {Ji, Feng and Ye, Shengkui},
title = {Vanishing of {L2{\textendash}Betti} numbers and failure of acylindrical hyperbolicity of matrix groups over rings},
journal = {Algebraic and Geometric Topology},
pages = {2825--2840},
publisher = {mathdoc},
volume = {17},
number = {5},
year = {2017},
doi = {10.2140/agt.2017.17.2825},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.2825/}
}
TY - JOUR AU - Ji, Feng AU - Ye, Shengkui TI - Vanishing of L2–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings JO - Algebraic and Geometric Topology PY - 2017 SP - 2825 EP - 2840 VL - 17 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.2825/ DO - 10.2140/agt.2017.17.2825 ID - 10_2140_agt_2017_17_2825 ER -
%0 Journal Article %A Ji, Feng %A Ye, Shengkui %T Vanishing of L2–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings %J Algebraic and Geometric Topology %D 2017 %P 2825-2840 %V 17 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.2825/ %R 10.2140/agt.2017.17.2825 %F 10_2140_agt_2017_17_2825
Ji, Feng; Ye, Shengkui. Vanishing of L2–Betti numbers and failure of acylindrical hyperbolicity of matrix groups over rings. Algebraic and Geometric Topology, Tome 17 (2017) no. 5, pp. 2825-2840. doi: 10.2140/agt.2017.17.2825
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