Homology of free quantum groups

TL;DR

This paper computes the Hochschild homology of free orthogonal quantum groups, establishes their Poincaré duality, and explores implications for $\\ell^2$-homology and free entropy dimension, revealing their 3-dimensional nature and vanishing Betti numbers.

Contribution

It provides the first Hochschild homology computation for free orthogonal quantum groups and links it to their geometric and analytical properties.

Findings

Hochschild homology satisfies Poincaré duality.

The quantum groups are considered 3-dimensional objects.

All $\\ell^2$ Betti numbers of $A_o(n)$ vanish.

Abstract

We compute the Hochschild homology of the free orthogonal quantum group $A_o(n)$. We show that it satisfies Poincar\'e duality and should be considered to be a 3-dimensional object. We then use recent results of R. Vergnioux to derive results about the $\ell^2$-homology of $A_o(n)$ and estimates on the free entropy dimension of its set of generators. In particular, we show that the $\ell^2$ Betti-numbers of $A_o(n)$ all vanish and that the free entropy dimension is less than 1.