# Word and Conjugacy Problems in Groups $G_{k+1}^{k}$

## TL;DR

This paper proves that the word and conjugacy problems for specific groups $G_{k+1}^k$, which generalize braid groups and relate to dynamical systems, are algorithmically solvable with constructive methods.

## Contribution

It establishes the algorithmic solvability of word and conjugacy problems for certain $G_{k+1}^k$ groups, advancing understanding of their algebraic structure.

## Key findings

- Word problem is algorithmically solvable for $G_{k+1}^k$ groups.
- Conjugacy problem is algorithmically solvable for $G_{k+1}^k$ groups.
- Constructive algorithms are provided for these problems.

## Abstract

Recently the third 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. In the present paper we prove that word and conjugacy problems for certain $G_{k+1}^k$ groups are algorithmically solvable, and the algorithms are constructive.