On the Two-Parabolic Subgroups of SL(2,C)

We consider homomorphisms $H_{t}$ from the free group $F$ of rank $2$ onto the subgroup of SL$(2,\mathbb{C})$ that is generated by two parabolic matrices. Up to conjugation, $H_{t}$ depends only on one complex parameter $t$. We study the possible relators, that is, the words $w\in F$ with $w\neq 1$ such that $H_{t}(w)=I$ for some $t\in\mathbb{C}$. We find several families of relators. Of particular interest here are relators connected with $2$-bridge knots, which we consider in a purely algebraic setting. We describe an algorithm to determine whether a given word is a possible relator.

On the Two-Parabolic Subgroups of SL(2,\mathbbC) Sobre los subgrupos dos-parabólicos de SL(2,\mathbbC) CHRISTIAN POMMERENKE1, MARGARITA TORO2

1Technische Universität Berlin, Berlin, Germany. Email: pommeren@math.tu-berlin.de 
2Universidad Nacional de Colombia, Medellín, Colombia. Email:mmtoro@unal.edu.co 

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Abstract

We consider homomorphisms Ht from the free group F of rank 2 onto the subgroup of SL(2,C) that is generated by two parabolic matrices. Up to conjugation, Ht depends only on one complex parameter t. We study the possible relators, that is, the words w∈ F with w≠ 1 such that Ht(w)=I for some t∈C. 
We find several families of relators. Of particular interest here are relators connected with 2-bridge knots, which we consider in a purely algebraic setting. We describe an algorithm to determine whether a given word is a possible relator.

Key words: Representation, Parabolic, Wirtinger presentation, Two-generated groups, Homomorphism, Longitude.

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2000 Mathematics Subject Classification: 15A30, 57M05.

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Resumen

Consideramos homomorfismos Ht del grupo libre F de rango 2 sobre el subgrupo de SL(2,C) que es generado por dos matrices parabólicas. Salvo conjugación, Ht depende sólo de un parámetro complejo t. Estudiamos los posibles relatores, esto es, las palabras w∈ F con w≠ 1 tal que Ht(w)=I para algún t∈C. 
Encontramos varias familias de relatores. De particular interés aquí son los relatores asociados con nudos de 2puentes, los cuales consideramos de forma puramente algebraica. Describimos un algoritmo para determinar cuándo una palabra dada es un posible relator.

Palabras clave: Representación, parabólico, presentación de Wirtinger, grupos dos-generados, homomorfismos, longitud.

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Texto completo disponible en PDF

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References

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(Recibido en septiembre de 2010. Aceptado en febrero de 2011)

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

@ARTICLE{RCMv45n1a04, 
    AUTHOR  = {Pommerenke, Christian and Toro, Margarita}, 
    TITLE   = {{On the Two-Parabolic Subgroups of SL\boldsymbol{(2,\mathbb{C})}}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2011}, 
    volume  = {45}, 
    number  = {1}, 
    pages   = {37-50} 
}