Jérémie BrieusselInstitut/Laboratoire Montpelliérain Alexander Grothendieck (IMAG) (UMR 5149), Université de Montpellier, 34090 Montpellier, FranceTianyi Zheng Department of Mathematics, Stanford University, Stanford (Palo Alto) CA 94305
Group Theory and Lie TheoryMetric GeometryProbabilitymathscidoc:2203.17002
We give a solution to the inverse problem (given a prescribed function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and Lp-compression functions of finitely generated groups. For smaller classes, we give solutions among solvable groups of exponential volume growth. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the Lp-compression exponent of a group and its wreath product with the cyclic group forp in [1,2].
Ilya GekhtmanDepartment of Mathematics, Technion-Israeli Institute of Technology, 32000 Haifa, IsraelVictor GerasimovDepartamento de Matemática, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Caixa Postal 702, 30161-970 BrasilLeonid PotyagailoUFR de Mathématiques, Université de Lille, 59655 Villeneuve d’Ascq, FranceWenyuan YangBeijing International Center for Mathematical Research, Peking University, Beijing 100871, China
Dynamical SystemsGeometric Analysis and Geometric TopologyGroup Theory and Lie TheoryProbabilitymathscidoc:2203.11005
For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.