On growth of random groups of intermediate growth

TL;DR

This paper investigates the growth patterns of typical groups within a specific family of intermediate growth p-groups, revealing oscillating growth behavior in a categorical sense and bounded growth almost surely.

Contribution

It provides a dual perspective on group growth, showing oscillating growth in a generic sense and bounded growth almost surely, highlighting complex growth dynamics.

Findings

Generic groups exhibit oscillating growth with no universal upper bound.

Almost surely, the growth function is bounded by $e^{n^eta}$ for some $eta<1$.

The study distinguishes between categorical and measure-theoretic growth behaviors.

Abstract

We study the growth of typical groups from the family of $p$-groups of intermediate growth constructed by the second author. We 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 $\alpha<1$.