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

TL;DR

This paper explores the algebraic groups $G_n^k$ and $\Gamma_n^k$, their connections to manifolds, dynamical systems, and invariants, providing a survey of recent developments in these interconnected mathematical areas.

Contribution

It introduces and reviews the foundational ideas and recent results related to the groups $G_n^k$ and $\Gamma_n^k$, highlighting their applications in topology, dynamics, and invariants.

Findings

$G_n^k$ groups relate to dynamical systems and invariants.

$\Gamma_n^k$ groups connect to manifold triangulations and topological invariants.

The paper surveys recent advances in the study of these groups and their applications.

Abstract

Recently the first named author defined a 2-parametric family of groups $G_n^k$. 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 1 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. 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 but there are many ways to enhance them to get rid of $2$-torsion. 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. Theorem of Pachner says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. $\Gamma_n^k$ naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing $\Gamma_n^k$: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a {\guillemotleft}braid group{\guillemotright} of the manifold and 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.