Metrics for quandles

A quandle is an algebraic system originating in knot theory, which can be regarded as a generalization of the operation of 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 these groups finitely generated, we investigate the graph structures induced by the actions, and the corresponding 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 of an important class of quandles, namely generalized Alexander quandles, has been studied in detail. We show that this metric space is quasi-isometric to the displacement group equipped with a word metric. Finally, we provide some examples that are quasi-isometric to typical metric spaces.