Browsing by Author "Pommerenke, Christian"

Now showing 1 - 5 of 5
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    Conjunto singular de familias analíticas de homomorfismos de grupos en PSL (2,C)
    (2013-07) Mejía Duque, Diego; Pommerenke, Christian; Toro Villegas, Margarita María
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    Grupos de nudos con dos generadores
    (2011-07) Pommerenke, Christian; Toro Villegas, Margarita María
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    Grupos de nudos con dos generadores
    (2011-05) Pommerenke, Christian; Toro Villegas, Margarita María
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    On the parametrized modular group
    (2011) Mejía Duque, Diego; Pommerenke, Christian; Toro Villegas, Margarita María
    Generalizing the modular and Hecke groups, we consider the subgroup ΙΙ of SL(2; Z[ξ]) generated by the parabolic (1/0 ξ/1) and the elliptic (0/1 (-1)/0)where Z[ξ] is the ring of polynomials in the variable ξ. For Ϛ Є C and W Є ΙΙ, W (Ϛ) means the matrix in SL (2, C) obtained when we evaluate the parameter ξ at Ϛ We enumerate the elements of ΙΙ and study the relators, defined as those W Є ΙΙ, for which there exists Ϛ Є C with W(ξ ) = ± I. Then, for W Є ΙΙ , we investigate the sets of for which W(Ϛ) is not loxodromic; their union is the singular set S(ΙΙ) C C. Its closure has been much studied for the two-parabolic group, which is a free subgroup of ΙΙ of index 4.
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    On the two-parabolic subgroups of SL (2; C)
    (2011) Pommerenke, Christian; Toro Villegas, Margarita María
    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 2 F with w 6= 1 such that Ht(w) = I for some t 2 C. We and several families of realtors. 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 and phrases. Representation, Parabolic, Wirtinger presentation, Two- generated groups, Homomorphism, Longitude.