Normal closure of one-variable equations in free groups
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- by C. Sibertin-Blanc PDF
- Proc. Amer. Math. Soc. 80 (1980), 34-38 Request permission
Abstract:
Let $w(x)$ be a one-variable equation in a free group F of finite rank. Lyndon has proved that it is possible to associate effectively to $w(x)$ the set of its solutions, whereas Appel and Lorenc have provided a simpler representation of the set inferred. In this paper, we invert the problem and demonstrate that if the elements of any set $S \subset F$ are solutions of an equation $w(x)$, then $w(x)$ belongs to the normal closure of finitely many short equations associated to S. A few consequences are given.References
- K. I. Appel, One-variable equations in free groups, Proc. Amer. Math. Soc. 19 (1968), 912–918. MR 232826, DOI 10.1090/S0002-9939-1968-0232826-3
- A. A. Lorenc, Representations of sets of solutions of systems of equations with one unknown in a free group, Dokl. Akad. Nauk SSSR 178 (1968), 290–292 (Russian). MR 0225861
- Roger C. Lyndon, Equations in free groups, Trans. Amer. Math. Soc. 96 (1960), 445–457. MR 151503, DOI 10.1090/S0002-9947-1960-0151503-8
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
- Arthur Steinberg, On equations in free groups, Michigan Math. J. 18 (1971), 87–95. MR 289614
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 80 (1980), 34-38
- MSC: Primary 20E05
- DOI: https://doi.org/10.1090/S0002-9939-1980-0574504-2
- MathSciNet review: 574504