H-convex standard fundamental domain of a subgroup of a modular group

Abstract

The two major ways of obtaining fundamental domains for discrete subgroups of SL(2,ℝ) are the Dirichlet Polygon construction (see Lehner in Discontinuous Groups and Automorphic Functions, American Mathematical Society, Providence, 1964) and Ford’s construction (see Ford in Automorphic Functions, McGraw–Hill, New York, 1929). Each of these two methods yield a hyperbolically convex fundamental domain for any discrete subgroup of SL(2,ℝ).

However, the Dirichlet polygon construction and Ford’s construction are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction and their reliance on knowing almost all elements of the group under discussion.

A third-and most important and practical-method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. Let Γ₂ be a subgroup of Γ₁ and

$$\Gamma_{1}=\Gamma_{2}\cdot \{L_{1},L_{2},\ldots,L_{m}\}.$$

If $\mathbb{F}$ is a fundamental domain of the bigger group Γ₁, then the set

$$\mathcal{R}_{\Gamma}=\Biggl(\overline{\bigcup_{k=1}^{m}L_{k}(\mathbb{F})}\,\Biggr)^{o}$$

(1)

is a fundamental domain of Γ₂. One can ask at this juncture, is it possible to choose the right cosets suitably so that the set ℛ_Γ is hyperbolically convex? We will answer this question affirmatively for

$$\Gamma_{1}=\Gamma(1)\quad \mbox{and}\quad \mathbb{F}=\biggl\{\tau \in \mathbb{H}:|\tau|>1\ \&\ |\mathrm{Re}(\tau)|<\frac{1}{2}\biggr\}.$$