On Groups $G_{n}^{k}$ and $Γ_{n}^{k}$: A Study of Manifolds, Dynamics, and Invariants

Vassily O. Manturov, Denis A. Fedoseev, Seongjeong Kim, Igor M. Nikonov

Published 2019-05-20Version 1

Recently the first named author defined a 2-parametric family of groups $G_n^k$ \cite{gnk}. Those groups may be regarded as a certain generalisation 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 and Kirillov-Fomin algebras, to name just a few. Study of the $G_n^k$ groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by $\Gamma_n^k$, 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~\cite{pach} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also~\cite{GelKapZel,nab}; the $\Gamma_n^k$ naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing this groups: 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 the present paper we give a survey of the ideas lying in the foundation of the $G_n^k$ and $\Gamma_n^k$ theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.