turkmath.org

The growth of a finitely generated group is an asymptotic invariant and measures the behaviour of sizes of balls in its Cayley graph. While examples of groups with polynomial and exponential growth are ubiquitous, the first examples of groups for which the growth is intermediate were constructed by R. Grigorchuk in 1984. We study the growth of typical groups from this family and find that, in the sense of category, a generic group exhibits oscillating growth with no universal upper bound. At the same time, from a measure-theoretic point of view (i.e., almost surely relative to an appropriately chosen probability measure), the growth function is bounded by $e^{n^\alpha}$ for some $\alpha<1$. This is joint work with R. Grigorchuk and Y. Vorobets.