Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 509-519

A GENERALIZATION OF THE PARAMETRIZED MODULAR GROUP

Christian Pommerenke and Margarita Toro

Technische Universität Berlin, Institut für Mathematik
D-10623 Berlin, Germany; pommeren 'at' math.tu-berlin.de

Universidad Nacional de Colombia, Sede Medellín, Escuela de Matemáticas
Calle 59a N. 63-20, Medellín, Colombia; mmtoro 'at' unal.edu.co

Abstract. Given a set of polynomials p₁,...,p_m ∈ C[ξ] we introduce the group Π = Π[p₁,...,p_m] = <A(p₁),...,A(p_m),B> where A(z) is the parabolic matrix (\begin{smallmatrix} 1&z \\ 0&1 \end{smallmatrix}) and B is the elliptic matrix (\begin{smallmatrix} 0&-1 \\ 1&0 \end{smallmatrix}). This group unifies the definitions of several groups that often appear in the literature. For instance, Π[1] is the modular group and Π[ξ] is the parametrized modular group introduced in [MPT15]. For m = 2, p₁ = 1, p₂ = i we have the Picard group Π[1,i] = SL(2,Z[i]). An important feature is the existence of a simple algorithm to obtain the elements of Π. We discuss several concrete examples, namely the euclidean Bianchi groups and a group from discrete relativity theory, furthermore the subgroup Π₁ of index 4 and its applications to knot theory.

2010 Mathematics Subject Classification: Primary 20G20, 15A30, 57M25, 83A05.

Key words: Parametrized modular group, Bianchi groups, knot groups, discrete relativity theory.

Reference to this article: Ch. Pommerenke and M. Toro: A generalization of the parametrized modular group. Ann. Acad. Sci. Fenn. Math. 43 (2018), 509-519.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4330