## Abstract

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right-angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most nonuniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (cocompact) right-angled arithmetic groups in SL(n,ℝ), n ≥ 3, and SO(p, q) for some values of p, q. This is a class of lattices for which the congruence subgroup property is not known in general. By using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right-angled lattice in a higher rank simple Lie group.

Original language | English (US) |
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Pages (from-to) | 2925-2964 |

Number of pages | 40 |

Journal | Duke Mathematical Journal |

Volume | 166 |

Issue number | 15 |

DOIs | |

State | Published - Oct 15 2017 |

### Funding

Abert's work is supported by a European Research Council Consolidator grant 648017 and a Magyar Tudományos Akadémia Lendület Groups and Graph Limits grant. Gelander's work is supported by Israel Science Foundation-Moked grant 2095/15. Nikolov's work is supported by Engineering and Physical Sciences Research Council grant EP/H045112/1 and the Clay Mathematical Institute.

## ASJC Scopus subject areas

- General Mathematics