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Mathematics > Group Theory
Title: On metric approximate subgroups
(Submitted on 2 Sep 2022 (v1), last revised 25 Apr 2024 (this version, v2))
Abstract: Let $G$ be a group with a metric $\mathrm{d}$ invariant under left and right translations, and let $\bar{\mathbb{D}}_r$ be the ball of radius $r$ around the identity. A $(k,r)$-metric approximate subgroup is a symmetric subset $X$ of $G$ such that the pairwise product set $XX$ is covered by at most $k$ translates of $X\bar{\mathbb{D}}_r$. This notion was introduced in arXiv:math/0601431 along with the version for discrete groups (approximate subgroups). In arXiv:0909.2190, it was shown for the discrete case that, at the asymptotic limit of $X$ finite but large, the "approximateness" (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on $X$ replacing finiteness. In particular, if $G$ has bounded exponent, we show that any $(k,r)$-metric approximate subgroup is close to a $(1,r')$-metric approximate subgroup for an appropriate $r'$.
Submission history
From: Arturo Rodríguez Fanlo [view email][v1] Fri, 2 Sep 2022 20:23:48 GMT (28kb)
[v2] Thu, 25 Apr 2024 18:59:37 GMT (30kb)
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