Metrics for quandles

TL;DR

This paper explores the algebraic and geometric structures of quandles, focusing on automorphism groups, Schreier graphs, and metrics, especially for generalized Alexander quandles, revealing their quasi-isometric properties.

Contribution

It introduces a detailed analysis of graph and metric structures induced by automorphism groups of finitely generated quandles, with new insights into generalized Alexander quandles.

Findings

Displacement group metrics are quasi-isometric to the associated metric spaces.

Schreier graphs generalize Cayley graphs for quandles.

Examples of quandles quasi-isometric to well-known metric spaces.

Abstract

A quandle is an algebraic system originating in knot theory, which can be regarded as a generalization of the conjugation of groups. This structure naturally defines two subgroups of its automorphism group, which are called the inner automorphism group and the displacement group, and they act on the quandle from the right. For a quandle with such groups being finitely generated, we investigate the graph structures induced from the actions, and induced metric spaces. The graph structures are defined by the notion of the Schreier graph, which is a natural generalization of the Cayley graph for a group. In particular, the metric associated with the displacement group for an important class of quandles, namely, generalized Alexander quandles, is studied in detail. We show that such a metric space is quasi-isometric to the displacement group with a word metric. Finally, we provide some examples quasi-isometric to typical metric spaces.