Metrics for quandles
Kohei Iwamoto, Ryoya Kai, Yuya Kodama
TL;DR
The paper develops a geometric framework forQuandles by endowing connected components with Schreier-graph–induced metrics coming from the inner automorphism group $\mathrm{Inn}(X)$ and the displacement group $\mathrm{Dis}(X)$. It proves that, when these acting groups are finitely generated, the resulting metric spaces have quasi-isometry classes that are invariant under choice of finite generating sets, and it identifies when displacement metrics coincide with word metrics in the free-action case. Special attention is given to generalized Alexander quandles $\mathrm{GAlex}(G,\sigma)$, where the displacement group is naturally isomorphic to the identity component of $G$, enabling a Milnor–Švarc–type correspondence between connected components and the corresponding groups. The work also presents explicit examples linking quandle geometry to classical spaces such as trees, Euclidean spaces, hyperbolic planes, and $3$-dimensional homogeneous spaces, thereby illustrating the breadth of possible quasi-isometric behaviors in quandle geometry.
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.