Tointon MCH. Approximate subgroups of residually nilpotent groups.
MATHEMATISCHE ANNALEN 2019;
374:499-515. [PMID:
31258186 PMCID:
PMC6560002 DOI:
10.1007/s00208-018-01795-z]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/18/2015] [Revised: 04/30/2018] [Indexed: 06/09/2023]
Abstract
We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there is an integer n > 1 such that the ball of radius n in some Cayley graph of G has cardinality bounded by n c log log n , then G is virtually ( log n ) -step nilpotent.
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