类灯群的分析、几何与度量方面
Analytic, geometric and measured aspects of lamplighter-like groups
- Université Paris Cité（巴黎西岱大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本论文从拟等距与度量角度研究类灯群，给出置换圈积的拟等距分类及刚性结果，并构造晕积的轨道等价耦合，改进等周轮廓估计。
中文摘要 AI 辅助
在本论文中，我们从多个视角研究接近圈积的有限生成群，这些群均定义为半直积。第一个主要结果是某些置换圈积的拟等距分类。我们所采用的策略依赖于Genevois和Tessera的最新工作，但产生了更强的刚性现象：在适当假设下，两个置换类灯群之间的拟等距总是诱导基群（配备相应正规子群）之间的对拟等距。我们还证明，在某些特定情况下，拟等距等价于双射拟等距。其中一章专门研究灯群具有多项式增长的标准圈积。我们建立了此类圈积之间拟等距的刚性结果：它们必须都是“度量缩放”的。特别地，此类圈积的自拟等距都位于距双射有界距离内。我们还证明了这类类灯群提供了可余拟等距群而非双射拟等距群的例子。论文的第二部分基于与Corentin Correia的合作工作，我们从度量观点研究Genevois和Tessera的晕积。我们构造了两个同类型晕积之间的轨道等价耦合，并利用Folner平铺的最新技术，展示了晕积与自由阿贝尔群之间的耦合。我们还建立了这些群等周轮廓的若干渐近估计，改进了Erschler-Zheng和Saloff-Coste-Zheng的先前结果。这些计算使我们能够证明，在许多情况下，由此获得的轨道等价耦合在数量上是最优的。
英文摘要
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.
