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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

Managing Editor: Khukhro, Evgenii I. / Kramer, Linus

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Solving one-variable equations in free groups

Dimitri Bormotov
  • Intrinsyc Software International, Inc., 700 W. Pender St., Vancouver, BC, V6C 1G8, Canada. E-mail:
/ Robert Gilman / Alexei Myasnikov
Published Online: 2008-11-06 | DOI: https://doi.org/10.1515/JGT.2008.080


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.

Received: 2007-06-22

Published Online: 2008-11-06

Published in Print: 2009-03-01

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

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