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 | 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.

**[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**, 10.1016/0021-8693(81)90265-9**[2]**Marc Culler,*Using surfaces to solve equations in free groups*, Topology**20**(1981), no. 2, 133–145. MR**605653**, 10.1016/0040-9383(81)90033-1**[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**, 10.1007/978-1-4613-9586-7_2**[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.

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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