2023/08/29 by Bruno Luiz Santos Correia, Marc Troyanov, Correia, Bruno Luiz Santos +1
Mathematics · #Geometric and Algebraic Topology #Mathematics and Applications #Advanced Combinatorial Mathematics
paper · pdf · doi:10.1007/978-3-031-43510-2_12
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 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-transform, including some computational techniques and its relation with the Legendre transform.