Orientation of quantum Cayley trees and applications

TL;DR

This paper introduces quantum Cayley graphs for quantum groups, focusing on quantum trees, and explores their properties, including orientations and applications to Property AO and K-theory, highlighting differences from classical cases.

Contribution

It defines quantum Cayley graphs for quantum groups, studies quantum tree orientations, and links these structures to Property AO and K-theory.

Findings

Quantum Cayley graphs generalize classical graphs to quantum groups.

Quantum ascending orientation leads to a non-involutive edge space at infinity.

Applications include insights into Property AO and K-theory for quantum groups.

Abstract

We introduce the quantum Cayley graphs associated to quantum discrete groups and study them in the case of trees. We focus in particular on the notion of quantum ascending orientation and describe the associated space of edges at infinity, which is an outcome of the non-involutivity of the edge-reversing operator and vanishes in the classical case. We end with applications to Property AO and K-theory.