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How to generalize known results on equations over groups. (English. Russian original) Zbl 1120.20033
Math. Notes 79, No. 3, 377-386 (2006); translation from Mat. Zametki 79, No. 3, 409-419 (2006).
A generalized equation over a group $$G$$ with variable group $$T$$ is a formal expression of the form $$g_1t_1g_2t_2\cdots g_nt_n=1$$, where $$g_i\in G$$ and $$t_i\in T$$. A generalized equation is said to be solvable over the group $$G$$ if there exists a group $$H$$ containing $$G$$ as a subgroup, and a homomorphism $$\mu\colon T\to H$$ for which $$g_1\mu(t_1)g_2\mu(t_2)\cdots g_n\mu(t_n)=1$$.
Known facts about the solvability of equations over groups are considered from this generalized point of view. In particular, it is shown that any unimodular generalized equation over a torsion-free group is solvable over this group. Here, unimodularity is understood in some special form that gives the usual unimodularity when the variable group $$T$$ is infinite cyclic. This result is obtained from a more general statement.

##### MSC:
 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F05 Generators, relations, and presentations of groups
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