Word and Conjugacy Problems in Groups $$\boldsymbol{G}_{\boldsymbol{k+1}}^{\boldsymbol{k}}$$

Recently the third named author defined a 2-parametric family of groups $$G_{n}^{k}$$ [12]. 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.