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15 October 2017 Rank, combinatorial cost, and homology torsion growth in higher rank lattices
Miklos Abert, Tsachik Gelander, Nikolay Nikolov
Duke Math. J. 166(15): 2925-2964 (15 October 2017). DOI: 10.1215/00127094-2017-0020

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), n3, 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.

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Miklos Abert. Tsachik Gelander. Nikolay Nikolov. "Rank, combinatorial cost, and homology torsion growth in higher rank lattices." Duke Math. J. 166 (15) 2925 - 2964, 15 October 2017. https://doi.org/10.1215/00127094-2017-0020

Information

Received: 8 October 2015; Revised: 12 September 2016; Published: 15 October 2017
First available in Project Euclid: 6 September 2017

zbMATH: 06812212
MathSciNet: MR3712168
Digital Object Identifier: 10.1215/00127094-2017-0020

Subjects:
Primary: 22E40
Secondary: 20F65

Keywords: arithmetic groups , cost , discrete subgroups of semisimple Lie groups , fixed price , higher-rank lattices , homology torsion , rank gradient

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 15 • 15 October 2017
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