Narrow Results

Recently, the first named author defined a 2-parametric family of groups $G_n^k$ [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 $G_n^k$ 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 $G_n^k$”.

The $G_n^k$ groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the $G_n^k$ groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the $G_n^k$ groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance $G_n^k$ 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 ${\mathrm{\Gamma }}_n^k$, 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 ${\mathrm{\Gamma }}_n^k$ naturally appear when considering the set of triangulations with the fixed number of points.

There are two ways of introducing the groups ${\mathrm{\Gamma }}_n^k$: 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 $G_n^k$ and ${\mathrm{\Gamma }}_n^k$ theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.