Krull dimension of groups and applications to large return probability

Lison Jacoboni (d'Orsay, U. Paris-Sud)

Wednesday 17th May, 2017 14:15-15:15 Maths 331B

Abstract

From the classical treatment of Krull dimension in ring theory, one can derive a notion of Krull dimension of a group. In this talk, I will present this notion and explain how it relates to the return probability of the random walk on a finitely generated group. For non-amenable groups and for groups of polynomial growth, the behaviour of the random walk is well-known thanks to works of Kesten and Varopoulos. Exponential growth groups only satisfy an upper bound in general. Many examples of groups reaching this bound are known - such as lamplighter groups, polycyclic groups, solvable groups of finite rank - as well as groups who do not. For metabelian groups, I will explain how this large behaviour can be characterized in terms of the Krull dimension. To understand this connection in the case of more general soluble groups, one need to know more about the structure of soluble groups of infinite rank, and I will describe a recent work with Peter Kropholler in this direction.

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