# The pro-$p$ group of upper unitriangular matrices

## TL;DR

This paper investigates the structure of the pro-$p$ group of upper unitriangular matrices, revealing subgroup embeddings, normalising properties, and the Hausdorff spectrum, linking it to $p$-adic groups and analytic structures.

## Contribution

It extends the subgroup analysis of the pro-$p$ group, embeds the Nottingham group, and characterizes the Hausdorff spectrum, providing new insights into its structure and connections.

## Key findings

- Embedding of the Nottingham group as a selfnormalising subgroup.
- Closure of Holubowski's free product is selfnormalising of infinite index.
- Hausdorff spectrum of the group is the entire interval [0,1].

## Abstract

We study the pro-$p$ group $G$ whose finite quotients give the prototypical Sylow $p$-subgroup of the general linear groups over a finite field of prime characteristic $p$. In this article, we extend the known results on the subgroup structure of $G$. In particular, we give an explicit embedding of the Nottingham group as a subgroup and show that it is selfnormalising. Holubowski (\cite{holub1,holub0,holub2}) studies a free product $C_p*C_p$ as a (discrete) subgroup of $G$ and we prove that its closure is selfnormalising of infinite index in the subgroup of $2$-periodic elements of $G$. We also discuss change of rings: field extensions and a variant for the $p$-adic integers, this latter linking $G$ with some well known $p$-adic analytic groups. Finally, we calculate the Hausdorff dimensions of some closed subgroups of $G$ and show that the Hausdorff spectrum of $G$ is the whole interval $[0,1]$ which is obtained by considering partition subgroups only.