zbMATH — the first resource for mathematics

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
Full Text:
References:
  S. D. Brodskii, ”Equations over groups and one-relator groups,” Sibirsk. Mat. Zh. [Siberian Math. J.], 25 (1984), no. 2, 84–103.  Ant. A. Klyachko and M. I. Prishchepov, ”The descent method for equations over groups,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1995), no. 4, 90–93. · Zbl 0914.20024  R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977; Russian translation: Mir, Moscow, 1980. · Zbl 0368.20023  A. Clifford, ”Nonamenable type K equations over groups,” Glasgow Math. J., 45 (2003), 389–400. · Zbl 1051.20011  A. Clifford and R. Z. Goldstein, ”Equations with torsion-free coefficients,” Proc. Edinburgh Math. Soc., 43 (2000), 295–307. · Zbl 0970.20019  M. M. Cohen and C. Rourke, ”The surjectivity problem for one-generator, one-relator extensions of torsion-free groups,” Geom. Topol., 5 (2001), 127–142. · Zbl 1014.20015  M. Edjvet and J. Howie, ”The solution of length four equations over groups,” Trans. Amer. Math. Soc., 326 (1991), 345–369. · Zbl 0737.20015  R. Fenn and C. Rourke, ”Klyachko’s methods and the solution of equations over torsion-free groups,” L’Enseignment Math., 42 (1996), 49–74. · Zbl 0861.20029  M. Gerstenhaber and O. S. Rothaus, ”The solution of sets of equations in groups,” Proc. Nat. Acad. Sci. USA, 48 (1962), 1531–1533. · Zbl 0112.02504  J. Howie, ”On pairs of 2-complexes and systems of equations over groups,” J. Reine Angew. Math., 324 (1981), 165–174. · Zbl 0447.20032  S. V. Ivanov and Ant. A. Klyachko, ”Solving equations of length at most six over torsion-free groups,” J. Group Theory, 3 (2000), 329–337. · Zbl 0971.20020  Ant. A. Klyachko, ”A funny property of a sphere and equations over groups,” Comm. Algebra, 21 (1993), 2555–2575. · Zbl 0788.20017  F. Levin, ”Solutions of equations over groups,” Bull. Amer. Math. Soc., 68 (1962), 603–604. · Zbl 0107.01803  R. C. Lyndon, ”Equations in groups,” Bol. Soc. Brasil. Math., 11 (1980), no. 1, 79–102. · Zbl 0463.20030  J. R. Stallings, ”A graph-theoretic lemma and group embeddings,” in: Combinatorial Group Theory and Topology (S. M. Gersten and J. R. Stallings, editors), Ann. of Math. Stud., vol. 111, 1987, pp. 145–155. · Zbl 0615.05034  S. D. Promyslow, ”A simple example of a torsion free nonunique product group,” Bull. London Math. Soc., 20 (1988), 302–304. · Zbl 0662.20022  E. Rips and Y. Segev, ”Torsion free groups without unique product property,” J. Algebra, 108 (1987), 116–126. · Zbl 0614.20021  A. Strojnowski, ”A note on u.p. groups,” Comm. Algebra, 8 (1980), 231–234. · Zbl 0423.20005  W. Magnus, ”Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz),” J. Reine Angew. Math., 163 (1930), 141–165. · JFM 56.0134.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.