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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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