# Topological finite generation of compact open subgroups of universal groups

## TL;DR

This paper characterizes when compact open subgroups of universal groups associated with certain permutation groups are topologically finitely generated, revealing they are also positively finitely generated in these cases.

## Contribution

It provides a characterization of permutation groups F for which all compact open subgroups of the associated universal group are topologically finitely generated.

## Key findings

- Identifies conditions on permutation groups F for topological finite generation
- Shows these groups are positively finitely generated
- Provides a classification related to universal groups and automorphisms of trees

## Abstract

In this paper we characterize the finite permutation groups $F<S_d$ on $d$ letters such that every compact open subgroup of the associated universal group $U(F)<{\rm Aut} T_d$ is topologically finitely generated. Actually we show that in this case the groups are positively finitely generated.