Abstract
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.
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ACKNOWLEDGMENTS
The authors are grateful to A.A. Klyachko and I.M. Nikonov for useful discussions.
Funding
The first named author was supported by the program ‘‘Leading Scientific Schools’’ (grant no. NSh-6399.2018.1, Agreement no. 075-02-2018-867) and by the Russian Foundation for Basic Research (grant no. 19-01-00775-a). The third named author was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (grant no. 14.Y26.31.0025 of the government of the Russian Federation).
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Fedoseev, D.A., Karpov, A.B. & Manturov, V.O. Word and Conjugacy Problems in Groups \(\boldsymbol{G}_{\boldsymbol{k+1}}^{\boldsymbol{k}}\). Lobachevskii J Math 41, 176–193 (2020). https://doi.org/10.1134/S1995080220020067
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DOI: https://doi.org/10.1134/S1995080220020067