# Random subgroups of branch groups

## TL;DR

This paper proves that independent Haar-random elements in super strongly fractal branch profinite groups generate free subgroups acting freely on the boundary, extending previous results to a broader class of groups.

## Contribution

It establishes a stronger free subgroup generation result for super strongly fractal branch profinite groups, improving upon earlier work that only showed almost free actions.

## Key findings

- Random elements generate free subgroups acting freely on the boundary
- Extension of previous results to super strongly fractal branch groups
- Generalization of earlier work on iterated wreath products

## Abstract

We show that independent Haar-random elements in a super strongly fractal branch profinite group generate a free subgroup acting freely on the boundary of the tree. This improves a previous result of Ab\'ert (2005) for weakly branch profinite groups, where independent random elements were shown to generate free subgroups acting only almost freely on the boundary. Our result also generalizes the analogous result of Ab\'ert and Vir\'ag (2005) for iterated wreath products.