Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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] Leo P. Comerford Jr. and Charles C. Edmunds, Quadratic equations over free groups and free products, J. Algebra 68 (1981), no. 2, 276–297. MR 608536,
  • [2] Marc Culler, Using surfaces to solve equations in free groups, Topology 20 (1981), no. 2, 133–145. MR 605653,
  • [3] S. M. Gersten, Reducible diagrams and equations over groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 15–73. MR 919828,
  • [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] Roger C. Lyndon, On the combinatorial Riemann-Hurwitz formula, Symposia Mathematica, Vol. XVII (Convegno sui Gruppi Infiniti, INDAM, Rome, 1973) Academic Press, London, 1976, pp. 435–439. MR 0409681
  • [7] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064
  • [8] S. J. Pride, Star complexes, (to appear in Glasgow J. Math.).
  • [9] C. P. Rourke, Presentations and the trivial group, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 134–143. MR 547460
  • [10] Paul E. Schupp, Quadratic equations in groups, cancellation diagrams on compact surfaces, and automorphisms of surface groups, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Stud. Logic Foundations Math., vol. 95, North-Holland, Amsterdam-New York, 1980, pp. 347–371. MR 579952
  • [11] H. B. Short, Topological methods in group theory: the adjunction problem, Ph.D. Thesis, Warwick, 1981.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 20F32, 20F06, 57M05, 57M20

Retrieve articles in all journals with MSC: 20F32, 20F06, 57M05, 57M20

Additional Information

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

American Mathematical Society