# On the two-parabolic subgroups of sl(2,c)

Pommerenke, Christian and Toro, Margarita (2011) On the two-parabolic subgroups of sl(2,c). Revista Colombiana de Matemáticas; Vol. 45, núm. 1 (2011); 37-50 0034-7426 .

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## Resumen

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.

Tipo de documento:Artículo - Article
Palabras clave:Representation, Parabolic, Wirtinger presentation, Two-generated groups, Homomorphism, Longitude, 15A30, 57M05