Author
Abert, M
Gelander, T
Nikolov, N
Journal title
Duke Mathematical Journal
DOI
10.1215/00127094-2017-0020
Issue
15
Volume
166
Last updated
2022-08-02T10:42:24.797+01:00
Page
2925-2964
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. <br/>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. <br/>Most non-uniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (co-compact) 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. <br/>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.
Symplectic ID
646375
Favourite
On
Publication type
Journal Article
Publication date
06 Sep 2016
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