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

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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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Authors and Affiliations

  1. Lomonosov Moscow State University, 119991, Moscow, Russia

    D. A. Fedoseev & A. B. Karpov

  2. V. A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 117997, Moscow, Russia

    D. A. Fedoseev

  3. Bauman Moscow State Technical University, 105005, Moscow, Russia

    V. O. Manturov

  4. Novosibirsk State University, 630090, Novosibirsk, Russia

    V. O. Manturov

Authors
  1. D. A. Fedoseev
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  2. A. B. Karpov
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  3. V. O. Manturov
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Correspondence to D. A. Fedoseev, A. B. Karpov or V. O. Manturov.

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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

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