Abstract
We study various aspects of the family of groups generated by the parabolic matrices A(t1 ζ), ... , A(tm ζ) where A(z) = ( 1 z0 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.
Highlights
A parabolic matrix is determined by one parameter
In this paper we study a family of groups generated by a finite number of parabolic matrices, where the parameter lies in a polynomial ring of one variable over the complex numbers
We studied a more general case when ξ runs through all complex numbers, so we introduce the parametrized modular group
Summary
In this paper we study a family of groups generated by a finite number of parabolic matrices, where the parameter lies in a polynomial ring of one variable over the complex numbers. We studied a more general case when ξ runs through all complex numbers, so we introduce the parametrized modular group. In the present paper we restrict ourselves to polynomials of the special form pk = tkξ where tk are complex numbers, we study the group. The closure of the singular set of analytic families of subgroups of PSL(2, C) has been much studied, see e.g. In order to establish the connections we needed to enlarge the groups Π(ζ) and consider groups generated by two or more parabolics This was a motivation to study the generalized parametrized modular group in [16].
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