On subgroup growth of iterated wreath products in product action

TL;DR

This paper investigates the subgroup growth of certain infinite groups constructed via iterated wreath products, demonstrating the existence of groups with a wide range of growth types between polynomial and quasi-polynomial.

Contribution

It introduces a method to compute subgroup growth types for hereditarily just infinite profinite groups formed through iterated wreath products, expanding understanding of subgroup growth spectrum.

Findings

Existence of hereditarily just infinite groups with subgroup growth between n and n^{log n}

Calculation of subgroup growth types for specific profinite groups

Demonstration of the diversity of subgroup growth behaviors in these groups

Abstract

We show that there are hereditarily just infinite groups of any subgroup growth type between $n$ and $n^{\log n}$. This is obtained calculating the subgroup growth type of a family of hereditarily just infinite profinite groups obtained via iterated wreath products of finite permutation groups with respect to product actions.