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arXiv · 2308.15376

Isoperimetry in Finitely Generated Groups

Bruno Luiz Santos Correia·Marc Troyanov
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Abstract

We revisit the isoperimetric inequalities for finitely generated groups introduced and studied by N. Varopoulos, T. Coulhon and L. Saloff-Coste. Namely we show that a lower bound on the isoperimetric quotient of finite subsets in a finitely generated group is given by the $\U-$transform of its growth function, which is a variant of the Legendre transform. From this lower bound, we obtain some asymptotic estimates for the F{\o}lner function of the group. The paper also includes a discussion of some basic definitions from Geometric Group Theory and some basic properties of the $\U$-transform, including some computational techniques and its relation with the Legendre transform.

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Bruno Luiz Santos Correia, Marc Troyanov. 2023-08-29. Isoperimetry in Finitely Generated Groups. https://doi.org/10.1007/978-3-031-43510-2_12

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