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.

中文翻译：

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组中的单词和共轭问题$$ \ boldsymbol {G} _ {\ boldsymbol {k + 1}} ^ {\ boldsymbol {k}} $$

摘要

最近，第三位具名的作者定义了一个由两参数组成的族\（G_ {n} ^ {k} \） [12]。这些组可以视为编织组的某种概括。对\（G_ {n} ^ {k} \）组与动力系统之间的联系的研究导致发现以下基本原理：``如果描述\（n \）粒子运动的动力系统拥有一个很好的余维一个属性恰好管辖\（K \）的颗粒，然后将这些动力系统承认拓扑不变量值在\（G_ {N} ^ {K} \） '' .The \（G_ {N} ^ {K} \ ）组与不同的代数结构有联系，例如Coxeter组和Kirillov-Fomin代数。研究\（G_ {n} ^ {k} \）组尤其导致不变量的构造，其在循环组的自由积中的价值。在本文中，我们证明了某些\（G_ {k +1} ^ {k} \）组在算法上可解决。