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Journal of Group Theory

Editor-in-Chief: Wilson, John S.

Managing Editor: Howie, James / Kramer, Linus / Parker, Christopher W.

Editorial Board Member: Abért, Miklós / Borovik, Alexandre V. / Boston, Nigel / Bridson, Martin R. / Caprace, Pierre-Emmanuel / Giovanni, Francesco / Guralnick, Robert / Jaikin Zapirain, Andrei / Kessar, Radha / Khukhro, Evgenii I. / Kochloukova, Dessislava H. / Malle, Gunter / Olshanskii, Alexander / Remy, Bertrand / Robinson, Derek J.S. / Willis, George

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Solving one-variable equations in free groups

Dimitri Bormotov1 / Robert Gilman2 / Alexei Myasnikov3

1Intrinsyc Software International, Inc., 700 W. Pender St., Vancouver, BC, V6C 1G8, Canada. E-mail:

2Stevens Institute of Technology, Hoboken NJ 07030-5991, U.S.A. E-mail:

3McGill University, Montreal, QC, H3A 2K6, Canada. E-mail:

Citation Information: Journal of Group Theory. Volume 12, Issue 2, Pages 317–330, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: 10.1515/JGT.2008.080, November 2008

Publication History

Published Online:


Equations in free groups have become prominent recently in connection with the solution to the well-known Tarski conjecture. Results of Makanin and Rasborov show that solvability of systems of equations is decidable and there is a method for writing down in principle all solutions. However, no practical method is known; the best estimate for the complexity of the decision procedure is P-space.

The special case of one-variable equations in free groups has been open for a number of years, although it is known that the solution sets admit simple descriptions. We use cancellation arguments to give a short and direct proof of this result and also to give a practical polynomial-time algorithm for finding solution sets. One-variable equations are the only general subclass of equations in free groups for which such results are known.

We improve on previous attempts to use cancellation arguments by employing a new method of reduction motivated by techniques from formal language theory. Our paper is self-contained; we assume only knowedge of basic facts about free groups.

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