# Large normal subgroup growth and large characteristic subgroup growth

## TL;DR

This paper investigates the growth of normal and characteristic subgroups in finitely generated groups, establishing conditions under which large subgroup growth types are inherited by larger groups, with implications for profinite groups and Golod-Shafarevich groups.

## Contribution

It proves that under certain conditions, groups with large subgroup growth inherit this property, extending previous results and providing new examples of groups with specific growth types.

## Key findings

- Normal subgroup growth of type n^{log n} can be inherited by larger groups in certain cases.
- Profinite groups with open subgroups that are Golod-Shafarevich have large normal subgroup growth.
- Existence of groups with characteristic subgroup growth of type n^{log n} is demonstrated.

## Abstract

The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\Gamma$ be a group and $\Delta$ a subgroup of finite index. Suppose $\Delta$ has normal subgroup growth of type $n^{\log n}$, does $\Gamma$ has normal subgroup growth of type $n^{\log n}$? We give a positive answer in some cases, generalizing a result of M\"uller and the second author and a result of Gerdau. For instance, suppose $G$ is a profinite group and $H$ an open subgroup of $G$. We show that if $H$ is a generalized Golod-Shafarevich group, then $G$ has normal subgroup growth of type of $n^{\log n}$. We also use our methods to show that one can find a group with characteristic subgroup growth of type $n^{\log n}$.