On groups G<inf>n</inf><inf>k</inf>and σ<inf>n</inf>^k: A study of manifolds, dynamics, and invariants

Abstract

Recently, the first named author defined a 2-parametric family of groups Gnk [V. O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups Gnk and dynamical systems led to the discovery of the following fundamental principle: "If dynamical systems describing the motion of n particles possess a nice codimension one property governed by exactly k particles, then these dynamical systems admit a topological invariant valued in Gnk". The Gnk groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the Gnk groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the Gnk groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance Gnk groups to get rid of this 2-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by σnk, which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin. 12(2) (1991) 129-145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math. 49(12) (1996) 1257-1270]; the σnk naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups σnk: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a "braid group"of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In this paper, we give a survey of the ideas lying in the foundation of the Gnk and σnk theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.