# Orthogonal free quantum group factors are strongly 1-bounded

## TL;DR

This paper proves that orthogonal free quantum group factors are strongly 1-bounded, distinguishing them from free group factors, by analyzing spectral regularity and applying free entropy dimension techniques.

## Contribution

It introduces a spectral regularity result for the edge reversing operator on the quantum Cayley tree and applies it to establish strong 1-boundedness of quantum group factors.

## Key findings

- Orthogonal free quantum group factors are strongly 1-bounded.
- These factors are not isomorphic to free group factors.
- Spectral regularity for the edge reversing operator is established.

## Abstract

We prove that the orthogonal free quantum group factors $\mathcal{L}(\mathbb{F}O_N)$ are strongly $1$-bounded in the sense of Jung. In particular, they are not isomorphic to free group factors. This result is obtained by establishing a spectral regularity result for the edge reversing operator on the quantum Cayley tree associated to $\mathbb{F}O_N$, and combining this result with a recent free entropy dimension rank theorem of Jung and Shlyakhtenko.