Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On metric approximate subgroups

E. Hrushovski, A. Rodriguez Fanlo

TL;DR

This work extends Hrushovski-style Lie model theory to metric groups by introducing a discretisation-based framework and model-theoretic notions of piecewise hyperdefinable sets. Through discretisation numbers, ultralimits, and compactified definable ideals, the authors construct a Metric Lie Model Theorem asserting that, under doubling-scale control and a constant-growth assumption, the ultraproduct of metric approximate subgroups admits a connected Lie model, with near-subgroup structure and strong commensurability relations to powers of the original set. The paper then derives corollaries that give explicit near-subgroup constructions and doubling-control outcomes, and finally connects these results to de Saxcé’s product theorem, producing product-type dichotomies for simple and semisimple Lie groups in the metric setting. Collectively, these contributions provide a robust structural theory for metric approximate subgroups, linking model theory, discretisation techniques, and Lie group geometry. The results offer a pathway to transfer finite approximate-subgroup structure into the metric regime, with potential applications to growth and regularity phenomena in non-discrete groups.

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 $r'$.

On metric approximate subgroups

TL;DR

This work extends Hrushovski-style Lie model theory to metric groups by introducing a discretisation-based framework and model-theoretic notions of piecewise hyperdefinable sets. Through discretisation numbers, ultralimits, and compactified definable ideals, the authors construct a Metric Lie Model Theorem asserting that, under doubling-scale control and a constant-growth assumption, the ultraproduct of metric approximate subgroups admits a connected Lie model, with near-subgroup structure and strong commensurability relations to powers of the original set. The paper then derives corollaries that give explicit near-subgroup constructions and doubling-control outcomes, and finally connects these results to de Saxcé’s product theorem, producing product-type dichotomies for simple and semisimple Lie groups in the metric setting. Collectively, these contributions provide a robust structural theory for metric approximate subgroups, linking model theory, discretisation techniques, and Lie group geometry. The results offer a pathway to transfer finite approximate-subgroup structure into the metric regime, with potential applications to growth and regularity phenomena in non-discrete groups.

Abstract

Let be a group with a metric invariant under left and right translations, and let be the ball of radius around the identity. A -metric approximate subgroup is a symmetric subset of such that the pairwise product set is covered by at most translates of . 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 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 replacing finiteness. In particular, if has bounded exponent, we show that any -metric approximate subgroup is close to a -metric approximate subgroup for an appropriate .
Paper Structure (11 sections, 26 theorems, 43 equations)

This paper contains 11 sections, 26 theorems, 43 equations.

Table of Contents

  1. Discretisations
  2. Preliminaries on piecewise hyperdefinable sets
  3. Building ideals
  4. Making functions definable
  5. Ultralimits
  6. Liminf of ideals
  7. Compactifications of ideals of definable sets
  8. Mapping ideals
  9. Metric Lie Model Theorem
  10. Applications
  11. Digression on de Saxcé product theorem

Key Result

Theorem 1

Let $(G_m,X_m,r_{i,m})_{i\leq m\in\mathbb{N}}$ be a sequence such that, for some fixed $k\in\mathbb{N}$, Let $(G^*,X,\ldots)$ be a non-principal ultraproduct with enough structure, $G$ the subgroup generated by $X$ and $o_r(1)=\bigcap_i \mathbb{D}_{r_i}(1)$, where $\mathbb{D}_{r_i}(1)$ is the ultraproduct of the balls of radius $r_{i,m}$. Then, $G\cdot o_r(1)\leq G^*$ has a connected Lie model $\

Theorems & Definitions (46)

  • Theorem 1: Metric Lie Model
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Lemma 5
  • Lemma 6
  • Remark 7
  • Lemma 8
  • Lemma 9
  • proof
  • ...and 36 more