Mark Sapir (Vanderbilt, Nashville, USA)

"Dimension growth of groups"

This is a joint talk with A. Dranishnikov.

Let G be a locally finite graph. Let a be a natural  number. Then we say that two vertices u, v are a-connected if there is a sequence of vertices u,u_1,...,u_n=v where the distance between consecutive vertices is at most a. For every coloring of vertices of G, by an a-cluster we call a maximal a-connected monochromatic set of vertices. Now let k=k(a) be the smallest number of colors such that there exists a k-coloring of G with a-clusters of bounded diameter. Then the function k(a) is called the dimension growth function of G. We study the growth rate of the function k(a) for finitely generated groups.