Commensurator Subgroups of Surface Groups

Let $M$ be a surface, and let $H$ be a subgroup of $\pi_{1}M$. In this paper we study the commensurator subgroup $C_{\pi_{1}M}(H)$ of $\pi_{1}M$, and we extend a result of L. Paris and D. Rolfsen \cite{Paris-Rolfsen}, when $H$ is a geometric subgroup of $\pi_{1}M$. We also give an application of commensurator subgroups to group representation theory. Finally, by considering certain closed curves on the Klein bottle, we apply a classification of these curves to self-intersection Nielsen theory.

Commensurator Subgroups of Surface Groups Subgrupos comensuradores del grupo fundamental de superfícies OSCAR EDUARDO OCAMPO URIBE1

1Universidade de São Paulo, São Paulo, Brasil. Email: oeocampo@ime.usp.br 

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Abstract

Let M be a surface, and let H be a subgroup of π1M. In this paper we study the commensurator subgroup C\\pi_1M(H) of π1M, and we extend a result of L. Paris and D. Rolfsen [7], when H is a geometric subgroup of π1M. We also give an application of commensurator subgroups to group representation theory. Finally, by considering certain closed curves on the Klein bottle, we apply a classification of these curves to self-intersection Nielsen theory.

Key words: Commensurator, Fundamental group, Surface.

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2000 Mathematics Subject Classification: 20F65, 57M05.

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Resumen

Sean M una superfície y H un subgrupo de π1M. En este artículo estudiamos los subgrupos conmensuradoresC\\pi_1M(H) de π1M, y extendemos un resultado obtenido por L. Paris y D. Rolfsen en [7], cuando H es un subgrupo geométrico de π1M. También daremos una aplicación de estos subgrupos conmensuradores a la teoría de representaciones de grupos. Finalmente, considerando ciertas curvas cerradas en la botella de Klein, aplicaremos una clasificación de estas curvas a la Teoría de Nielsen de auto-intersección.

Palabras clave: Comensurador, grupo fundamental, superfície.

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Texto completo disponible en PDF

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References

[1] S. A. Bogatyi, E. A. Kudryavtseva, and H. Zieschang, `On the Coincidence Points of Mappings of a Torus Into a Surface´, (Russian. Russian summary) Tr. Mat. Inst. Steklova 247, (2004), 15-34. Geom. Topol. i Teor. Mnozh, translation in Proc. Steklov Inst. Math. 2004, no. 4 (247), 9-27

[2] M. Burger and P. d. l. Harpe, `Constructing Irreducible Representations of Discrete Groups´, Proc. Indian Acad. Sci. Math. Sci. 107, 3 (1997), 223-235.

[3] D. R. J. Chillingworth, `Winding Numbers on Surfaces. II´, Math. Ann. 199, (1972), 131-153.

[4] H. B. Griffiths, `The Fundamental Group of a Surface, and a Theorem of Schreier´, Acta Math. 110, (1963), 1-17.

[5] G. W. Mackey, The Theory of Unitary Group Representations, University of Chicago Press, 1976.

[6] O. E. Ocampo, Subgrupos geométricos e seus comensuradores em grupos de tranças de superfície, Dissertação de Mestrado, Universidade de São Paulo, São Paulo, Brasil, 2009.

[7] L. Paris and D. Rolfsen, `Geometric Subgroups of Surface Braid Groups´, Ann. Inst. Fourier 49, (1999), 417-472.

[8] D. Rolfsen, `Braid Subgroup Normalisers, Commensurators and Induced Representations´, Invent. Math. 68, (1997), 575-587.

[9] G. P. Scott, `Subgroups of Surface Groups are almost Geometric´, J. London Math. Soc. 17, (1978), 555-565.

(Recibido en junio de 2009. Aceptado en mayo de 2010)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv44n1a01, 
    AUTHOR  = {Ocampo Uribe, Oscar Eduardo}, 
    TITLE   = {{Commensurator Subgroups of Surface Groups}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2010}, 
    volume  = {44}, 
    number  = {1}, 
    pages   = {1-13} 
}