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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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?)

  • [1] L. P. Comerford, Jr., and C. C. Edmonds, Quadratic equations over free groups and free products, J. Algebra 68 (1981), 276-297. MR 608536 (82k:20060)
  • [2] M. Culler, Using surfaces to solve equations in groups, Topology 20 (1981), 133-145. MR 605653 (82c:20052)
  • [3] S. M. Gersten, Reducible diagrams and equations over groups, preprint. MR 919828 (89d:20030)
  • [4] R. Z. Goldstein and E. C. Turner, Solving quadratic equations in groups, preprint.
  • [5] Z-B. Gu, Cancellation diagrams on surfaces and quadratic equations in groups, (to be submitted).
  • [6] R. C. Lyndon, On the combinatorial Riemann-Hurwitz formula, in Convegni sui gruppi infiniti. Rome 1973, Academic Press, New York and London, 1976, 435-439. MR 0409681 (53:13433)
  • [7] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Ergebnisse der math. 89, Springer-Verlag, 1977. MR 0577064 (58:28182)
  • [8] S. J. Pride, Star complexes, (to appear in Glasgow J. Math.).
  • [9] C. Rourke, Presentations and the trivial group, in Topology of Low-Dimensional Manifolds, Lecture Notes in Math. Vol. 722, Springer-Verlag, 1979, 134-143. MR 547460 (81a:57001)
  • [10] P. E. Schupp, Quadratic equations in groups, cancellation diagrams on compact surfaces and automorphisms of surface groups, in Word Problems II: The Oxford Book, Amsterdam, 1980, 347-371. MR 579952 (81i:20040)
  • [11] H. B. Short, Topological methods in group theory: the adjunction problem, Ph.D. Thesis, Warwick, 1981.

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Keywords: Quadratic relations, cancellation diagrams, groups of hyperbolic $ 2$-complexes, free and nonfree solutions, hyperbolic surfaces
Article copyright: © Copyright 1989 American Mathematical Society

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