ON RANDOM GROUPS OF INTERMEDIATE GROWTH

In 1968 it became apparent that all known classes of groups have either polynomial or exponential growth, and John Milnor formally asked whether groups of intermediate growth exist. In 1984, the speaker introduced the first such examples and, in fact, constructed a continuum of groups with different intermediate growth, so, in particular, these groups are pairwise non-quasi-isometric. This continuum can be viewed as a Cantor subset X in the space of marked groups with a natural continuous map T: X ----> X which preserves many group properties. In fact, the dynamical system (T,X) is topologically conjugate to the one sided shift. I will explain why, for any reasonable T-invariant probability measure on X, a typical (i.e. -almost) property of a group from the family of groups X is to have growth bounded from above by a function of the type exp{n^\alpha}, where \alpha is a constant smaller that 1. At the same time, a co-meager subset Y of X of groups has a different type of behavior at infinity, namely it has the so-called oscillating growth which I will define during the talk. At the beginning of the talk I will also discuss a general approach to randomness in group theory based on the use of the space of marked groups.