Groups, Geometry, and Dynamics

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Volume 5, Issue 2, 2011, pp. 213–230
DOI: 10.4171/GGD/124

The rank gradient from a combinatorial viewpoint

Miklós Abért[1], Andrei Jaikin-Zapirain[2] and Nikolay Nikolov[3]

(1) Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364, BUDAPEST, HUNGARY
(2) Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049, MADRID, SPAIN
(3) Mathematical Institute, University of Oxford, Andrew Wiles Bldg, Woodstock Road, OX2 6GG, OXFORD, UNITED KINGDOM

This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the amenable case we generalize Lackenby's trichotomy theorem on finitely presented groups.

Keywords: Rank gradient, amalgam, amenable group, Lück approximation

Abért Miklós, Jaikin-Zapirain Andrei, Nikolov Nikolay: The rank gradient from a combinatorial viewpoint. Groups Geom. Dyn. 5 (2011), 213-230. doi: 10.4171/GGD/124