Growth estimates for discrete quantum groups

TL;DR

This paper explores growth properties of discrete quantum groups, providing general insights, explicit calculations, and verifying quantum analogues of classical growth estimates for duals of Lie groups.

Contribution

It introduces the notion of growth for discrete quantum groups and verifies quantum Gromov's estimate for duals of classical Lie groups.

Findings

Established the concept of growth in discrete quantum groups.

Verified quantum Gromov's estimate for duals of classical Lie groups.

Provided explicit computations illustrating growth behaviors.

Abstract

We discuss the notion of growth for discrete quantum groups, with a number of general considerations, and with some explicit computations. Of particular interest is the quantum analogue of Gromov's estimate regarding polynomial growth: we formulate the precise question, and we verify it for the duals of classical Lie groups.