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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.22 no.2 Antofagasta Aug. 2003

http://dx.doi.org/10.4067/S0716-09172003000200002 

Proyecciones
Vol. 22, N o 2, pp. 117-125, August 2003.
Universidad Católica del Norte
Antofagasta - Chile

A COMMUTATOR RIGIDITY FOR
FUNCTION GROUPS AND TORELLI’S
THEOREM*

RUBÉN HIDALGO
Universidad Téecnica Federico Santa María, Valparaíso – Chile

Abstract

We show that a non-elementary finitely generated torsion-free func-tion group is uniquely determined by its commutator subgroup. In this
way, we obtain a generalization of the results obtained in [2], [3] and
[8]. This is well related to Torelli’s theorem for closed Riemann sur-faces.For a general non-elementary torsion-free Kleinian group the
above rigidity property still unknown.

2000 Mathematics Subject Classification. Primary 30F40.

Key words and phrases. Kleinian groups, Function groups,
Torelli’s theorem, Hyperbolic 3-manifolds.


References

[1] H. Farkas and I. Kra. Riemann surfaces. Graduate Texts in Mathe-matics, Springer-Verlag.        [ Links ]

[2] R. Hidalgo. Homology coverings of Riemann surfaces, Tˆohoku Math.J. 45 (1993), 499-503.        [ Links ]

[3] R. Hidalgo, Kleinian groups with common commutator subgroup,Complex variables 28, pp. 121-133, (1995).        [ Links ]

[4]R. Hidalgo. Noded Fuchsian groups, Complex Variables 36, pp. 45-66, (1998).        [ Links ]

[5] R. Hidalgo. Noded function groups. Contemporary Mathematics. 240, pp. 209-222, (1999).        [ Links ]

[6] R. Hidalgo. Homology covering of closed Klein surfaces. Revista Proyecciones 18, pp. 165-173, (1999).        [ Links ]

[7] R. Hidalgo. A note on the homology covering of analytically finite Klein surfaces. Complex variables 42, pp. 183-192, (2000).        [ Links ]

[8] B. Maskit. The homology covering of a Riemann surface, Tôhoku Math. J. 38, pp. 561-562, (1986).        [ Links ]

[9] B. Markit. Kleinian Groups Grundlehren der Mathematischen Wis-senschaften, Vol. 287, Springer-Verlag, (1988).        [ Links ]

[10] B. Maskit. On boundaries of Teichmüller spaces and on kleinian groups II, Ann. of Math. 91, pp. 607-639, (1970).        [ Links ]Received : March 2003.

Rubén Hidalgo
Departamento de Matemática
Universidad Técnica Federico Santa María
Casilla 110 - V
Valparaíso
Chile.
E-mail: rhidalgo@mat.utfsm.cl


*This work was partially supported by projects UTFSM 12.03.21 and Fondecyt 1030252, 1030373.

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