Word and Conjugacy Problems in Groups G^k_k+1

Recently the third named author defined a 2-parametric family of groups G^k_n [12]. Those groups may be regarded as a certain generalisation of braid groups. Study of the connection between the groups G^k_n 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^k_n. The G^k_n groups have connections to different algebraic structures, Coxeter groups and Kirillov–Fomin algebras, to name just a few. Study of the G^k_n 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_k+1 groups are algorithmically solvable.