Analytic, geometric and measured aspects of lamplighter-like groups

Vincent Dumoncel

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arXiv · 2609.33756

Analytic, geometric and measured aspects of lamplighter-like groups

Vincent Dumoncel

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Abstract

In this thesis, we study finitely generated groups close to wreath products, all defined as semi-direct products, from various perspectives. The first main result is a quasi-isometric classification of some permutational wreath products. The strategy we follow relies on recent work of Genevois and Tessera, but yields a stronger rigidity phenomenon: under suitable assumptions, a quasi-isometry between two permutational lamplighters always induces a quasi-isometry of pairs between the base groups equipped with the corresponding normal subgroups. We also show that, in some specific cases, being quasi-isometric is equivalent to being bijectively quasi-isometric. A chapter is also dedicated to the study of standard wreath products whose lamp groups have polynomial growth. We establish a rigidity result for quasi-isometries between such wreath products: they must all be «measure-scaling». In particular, self-quasi-isometries of such wreath products all lie at bounded distance from bijections. We also prove that this class of lamplighters provides examples of amenable quasi-isometric groups that are not bijectively quasi-isometric. The second part of this thesis relies on joint works with Corentin Correia, in which we study halo products of Genevois and Tessera from a measured point of view. We construct orbit equivalence couplings between two halo products of the same kind and, using the recent technology of Folner tilings, we exhibit couplings between halo products and free abelian groups. We also establish several asymptotic estimates of the isoperimetric profiles of these groups, improving previous results of Erschler-Zheng and Saloff-Coste-Zheng. These computations allow us to prove that, in many cases, the orbit equivalence couplings thus obtained are quantitatively optimal.

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Vincent Dumoncel. 2026-09-27. Analytic, geometric and measured aspects of lamplighter-like groups. https://arxiv.org/abs/2609.33756

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