On a family of groups generated by parabolic matrices

Christian Pommerenke; Margarita Toro

doi:10.15446/recolma.v53n2.85541

Publicado

2019-07-01

On a family of groups generated by parabolic matrices

Sobre una familia de grupos generados por matrices parabólicas

DOI:

https://doi.org/10.15446/recolma.v53n2.85541

Palabras clave:

modular group, parametrized modular group, singular set, discrete groups, Chebyshev polynomials (en)
grupo modular, grupo modular parametrizado, conjunto singular, grupos discretos, polinomios de Chebyshev (es)

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Autores/as

Christian Pommerenke Technische Universität Berlin
Margarita Toro Universidad Nacional de Colombia

Resumen (en)
Resumen (es)

We study various aspects of the family of groups generated by the parabolic matrices A(t₁ ζ), ... , A(t_m ζ) where A(z) = ( 1 z
0 1 ) and by the elliptic matrix ( 0 -1 1 0 ). The elements of the matrices W in such groups can be computed by a recursion formula. These groups are special cases of the generalized parametrized modular groups introduced in [16].
We study the sets {z : tr W(z) ∈ [-2; +2]} [13] and their critical points and geometry, furthermore some finite index subgroups and the discretness of subgroups.

Estudiamos algunos aspectos de la familia de grupos generados por matrices parabólicas A(t₁ ζ), ... , A(t_m ζ) donde A(z) = ( 1 z
0 1 ) y por la matriz elíptica ( 0 -1 1 0 ). Los elementos de las matrices W en tales grupos se pueden calcular mediante una fórmula de recurrencia. Estos grupos son casos especiales de la generalización del grupo modular parametrizado estudiado en [16].
Estudiamos los conjuntos {z : tr W(z) ∈ [-2; +2]} [13] y sus puntos críticos y geometría, así como también algunos subgrupos de índice finito y la discreticidad de tales subgrupos.

Referencias

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Cómo citar

APA

Pommerenke, C. y Toro, M. (2019). On a family of groups generated by parabolic matrices. Revista Colombiana de Matemáticas, 53(2), 221–236. https://doi.org/10.15446/recolma.v53n2.85541

ACM

[1]

Pommerenke, C. y Toro, M. 2019. On a family of groups generated by parabolic matrices. Revista Colombiana de Matemáticas. 53, 2 (jul. 2019), 221–236. DOI:https://doi.org/10.15446/recolma.v53n2.85541.

ACS

(1)

Pommerenke, C.; Toro, M. On a family of groups generated by parabolic matrices. rev.colomb.mat 2019, 53, 221-236.

ABNT

POMMERENKE, C.; TORO, M. On a family of groups generated by parabolic matrices. Revista Colombiana de Matemáticas, [S. l.], v. 53, n. 2, p. 221–236, 2019. DOI: 10.15446/recolma.v53n2.85541. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/85541. Acesso em: 28 ene. 2025.

Chicago

Pommerenke, Christian, y Margarita Toro. 2019. «On a family of groups generated by parabolic matrices». Revista Colombiana De Matemáticas 53 (2):221-36. https://doi.org/10.15446/recolma.v53n2.85541.

Harvard

Pommerenke, C. y Toro, M. (2019) «On a family of groups generated by parabolic matrices», Revista Colombiana de Matemáticas, 53(2), pp. 221–236. doi: 10.15446/recolma.v53n2.85541.

IEEE

[1]

C. Pommerenke y M. Toro, «On a family of groups generated by parabolic matrices», rev.colomb.mat, vol. 53, n.º 2, pp. 221–236, jul. 2019.

MLA

Pommerenke, C., y M. Toro. «On a family of groups generated by parabolic matrices». Revista Colombiana de Matemáticas, vol. 53, n.º 2, julio de 2019, pp. 221-36, doi:10.15446/recolma.v53n2.85541.

Turabian

Pommerenke, Christian, y Margarita Toro. «On a family of groups generated by parabolic matrices». Revista Colombiana de Matemáticas 53, no. 2 (julio 1, 2019): 221–236. Accedido enero 28, 2025. https://revistas.unal.edu.co/index.php/recolma/article/view/85541.

Vancouver

1.

Pommerenke C, Toro M. On a family of groups generated by parabolic matrices. rev.colomb.mat [Internet]. 1 de julio de 2019 [citado 28 de enero de 2025];53(2):221-36. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/85541