Isoperimetry in Finitely Generated Groups

Bruno Luiz Santos Correia () and Marc Troyanov ()
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Bruno Luiz Santos Correia: Institut de Mathématiques EPFL
Marc Troyanov: Institut de Mathématiques EPFL

Chapter Chapter 12 in Surveys in Geometry II, 2024, pp 361-386 from Springer

Abstract: 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 − $$\mathcal {U}-$$ transform of its growth function, which is a variant of the Legendre transform. From this lower bound, we obtain some asymptotic estimates for the Følner 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 $$\mathcal {U}$$ -transform, including some computational techniques and its relation with the Legendre transform.

Keywords: Finitely generated groups; Growth; Isoperimetric inequality; Følner function; Lambert W-function; 20F65; 20F69 (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-43510-2_12

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DOI: 10.1007/978-3-031-43510-2_12

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