## 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 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_{2n}(R), the elementary symplectic group ESp_{2n}(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.

Original language | English |
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Pages (from-to) | 2825-2840 |

Number of pages | 16 |

Journal | Algebraic and Geometric Topology |

Volume | 17 |

Issue number | 5 |

DOIs | |

Publication status | Published - 19 Sept 2017 |

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