We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple gluing of such. In one of our simplest constructions—the pocket-extension of a group G—this leads to the study of certain finitely generated subgroups of the full permutation group \({\mathbb {S}}(G\cup \{*\})\). Some sharp estimates are obtained while many challenging questions remain.

中文翻译：

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由分段动作定义的某些组的等周曲线和随机游走

我们研究了通过对标记 Schreier 图的操作和对其进行简单粘合而定义的有限生成组的某些族的等周和谱图。在我们最简单的构造之一——群G的口袋扩展——这导致了对全置换群\({\mathbb {S}}(G\cup \{*\}) \)。获得了一些尖锐的估计，但仍然存在许多具有挑战性的问题。