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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
MathSciNet review: 975644
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Abstract: A group of a hyperbolic $ 2$-complex $ K$ 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 $ ({y_1}, \ldots ,{y_n}) = 1$ in $ K$, where $ W$ is a path in a $ 2$-complex $ B$, is a mapping $ \zeta :B \to K$ such that $ \zeta W = W(\zeta {y_1}, \ldots ,\zeta {y_n})$ is contractible in $ K$. This solution $ \zeta $ is free if there is a mapping $ h:B \to {K^{(1)}}$ such that $ W(h{y_1}, \ldots ,h{y_n})$ is contractible in $ {K^{(1)}}$ and such that $ \zeta = \pi h$, where $ \pi $ is the projection $ \pi :{K^{(1)}} \to K$. Our main result is that each quadratic equation $ W = 1$ has only finitely many nonfree solutions in $ K$. Our tool is essentially the cancellation diagrams on surfaces developed by the present author based on work of Schupp.


References [Enhancements On Off] (What's this?)

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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 $ 2$-complexes, free and nonfree solutions, hyperbolic surfaces
Article copyright: © Copyright 1989 American Mathematical Society