# Topological generation results for free unitary and orthogonal groups

## TL;DR

This paper proves that free unitary and orthogonal quantum groups are topologically generated by classical groups and lower-rank quantum groups, leading to new results on their finiteness properties and residual finiteness.

## Contribution

It establishes a uniform inductive approach to show that free quantum groups are topologically generated by classical groups and lower-rank quantum groups, revealing new finiteness properties.

## Key findings

- Free unitary groups are topologically generated by classical unitary groups and lower-rank free unitary quantum groups.
- All discrete quantum duals of these groups are residually finite, implying the Kirchberg factorization property.
- The groups are also shown to be hyperlinear and generated by their maximal tori.

## Abstract

We show that for every $N\ge 3$ the free unitary group $U^+_N$ is topologically generated by its classical counterpart $U_N$ and the lower-rank $U^+_{N-1}$. This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all $N\ne 3$, also hold at $N=3$. Specifically, all discrete quantum duals $\widehat{U^+_N}$ and $\widehat{O^+_N}$ are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, $U^+_N$ are topologically generated by $U_N$ and their maximal tori $\widehat{\mathbb{Z}^{*N}}$ (dual to the free groups on $N$ generators) and similarly, $O^+_N$ are topologically generated by $O_N$ and their tori $\widehat{\mathbb{Z}_2^{*N}}$.