Portrait growth in contracting, regular branch groups

TL;DR

This paper derives recursive formulas and asymptotic behavior for the portrait growth of the first Grigorchuk group and other related groups, advancing understanding of their subgroup structures and growth patterns.

Contribution

It introduces a general recursive framework for portrait growth in finitely generated, contracting, regular branch groups, including specific groups like GGS-groups and the Apollonian group.

Findings

Recursive formulas for portrait growth of the first Grigorchuk group.

Asymptotic results for portrait growth in these groups.

Complete description of portrait growth for GGS-groups and the Apollonian group.

Abstract

We address a question of Grigorchuk by providing both a system of recursive formulas and an asymptotic result for the portrait growth of the first Grigorchuk group. The results are obtained through analysis of some features of the branching subgroup structure of the group. More generally, we provide recursive formulas for the portrait growth of any finitely generated, contracting, regular branch group, based on the coset decomposition of the groups that are higher in the branching subgroup structure in terms of the lower subgroups. Using the same general approach we fully describe the portrait growth for all non-symmetric GGS-groups and for the Apollonian group.