$n$-valued groups, branched coverings and hyperbolic 3-manifolds

V. M. Buchstaber ; A. Yu. Vesnin

Sbornik. Mathematics, Tome 215 (2024) no. 11, pp. 1441-1467

Résumé

The theory of $n$-valued groups and its applications is developed by going over from groups defined axiomatically to combinatorial groups defined by generators and relations. A wide class of cyclic $n$-valued groups is introduced on the basis of cyclically presented groups. The best-known cyclically presented groups are the Fibonacci groups introduced by Conway. The problem of the existence of the orbit space of $n$-valued groups is related to the problem of the integrability of $n$-valued dynamics. Conditions for the existence of such spaces are presented. Actions of cyclic $n$-valued groups on $\mathbb R^3$ with orbit space homeomorphic to $S^3$ are constructed. The projections $\mathbb R^3 \to S^3$ onto the orbit space are shown to be connected, by means of commutative diagrams, with coverings of the sphere $S^3$ by three-dimensional compact hyperbolic manifolds which are cyclically branched along a hyperbolic knot. Bibliography: 54 titles.

Détail

Keywords: $n$-valued group, cyclically presented group, branched cyclic covering, three-dimensional manifold, knot.
Mots-clés : Fibonacci group