Hyperbolic surfaces and quadratic equations in groups

Author:
Zhi-Bin Gu

Journal:
Proc. Amer. Math. Soc. **107** (1989), 859-866

MSC:
Primary 20F32; Secondary 20F06, 57M05, 57M20

DOI:
https://doi.org/10.1090/S0002-9939-1989-0975644-4

MathSciNet review:
975644

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Abstract | References | Similar Articles | Additional Information

Abstract: A group of a hyperbolic -complex is a group with its associated van Kampen diagrams satisfying a hyperbolic curvature condition and a link condition on the degree of the interior vertices. *A solution* of an equation in , where is a path in a -complex , is a mapping such that is contractible in . This solution is *free* if there is a mapping such that is contractible in and such that , where is the projection . Our main result is that each quadratic equation has only finitely many nonfree solutions in . Our tool is essentially the cancellation diagrams on surfaces developed by the present author based on work of Schupp.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0975644-4

Keywords:
Quadratic relations,
cancellation diagrams,
groups of hyperbolic -complexes,
free and nonfree solutions,
hyperbolic surfaces

Article copyright:
© Copyright 1989
American Mathematical Society