A family of groups $G_{n}^{k}$ may be regarded as analogues of braid groups

and Coxeter groups. Defining relations for look like group-like versions

of the ``tetrahedron relations'' (higher Yang-Baxter-Relations).

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 $G_{n}^{k}$.}

These groups have connections to different algebraic structures;

in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the groups are reflections but there are many ways to enhance them to get rid of $2$-torsion. Later

joint work with I.M.Nikonov introduced and studied a second family of groups, which are closely related to sets of triangulations of manifolds fixed number of points.

There are two ways of introducing these groups: geometrical (depending on the metric) and topological. The second one can be thought of as a ``braid group'' of the manifold and is an invariant of the topological type of the manifold; in a similar way, one can construct the smooth version. The talk is a survey giving an overview of recent results

related to manifolds, dynamical systems, knots and braids.

Many problems and research projects will be formulated at the end of the talk.

V.O. Manturov, D.A. Fedoseev, S. Kim, I.M. Nikonov, “On groups Gkn and Γkn: A study of manifolds, dynamics, and invariants”, *Bulletin of Mathematical Sciences*, **11**:02 (2021), 2150004

{Book} V.O.Manturov, D.A.Fedoseev, S.Kim, and I.M.Nikonov,

Invariants and Pictures:Low-Dimensional Topology and Combinatorial Group Theory,