On Groups G_n^k and Γ_n^k: A Study of Manifolds, Dynamics, and Invariants

Manturov, V.O.; Fedoseev, D.A.; Kim, S.; Nikonov, I.M.

Авторы: Manturov Vassily O., Fedoseev Denis A., Kim Seongjeong, Nikonov Igor M.
Журнал: BULLETIN OF MATHEMATICAL SCIENCES
Год издания: 2021
Номер статьи: S1664360721500041
DOI: 10.1142/s1664360721500041
Аннотация: Recently the first named author defined a 2-parametric family of groups Gkn [31]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups Gkn 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 Gkn”. The Gkn groups have connections to different algebraic structures, Coxeter groups, Kirillov- Fomin algebras, and cluster algebras, to name three. Study of the Gkn groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the Gkn groups are reflections which makes them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance Gkn 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 Γkn, 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 [51] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [14, 47]; the Γkn naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups Γkn: 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 Gkn and Γkn theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.
Добавил в систему: Федосеев Денис Александрович

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On Groups G_n^k and Γ_n^k: A Study of Manifolds, Dynamics, and Invariantsстатья Исследовательская статья

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