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\(H\)-convex standard fundamental domain of a subgroup of a modular group. (English) Zbl 1172.30016
Summary: The two major ways of obtaining fundamental domains for discrete subgroups of \(\text{SL}(2,\mathbb R)\) are the Dirichlet polygon construction (see J. Lehner in [Discontinuous groups and automorphic functions, American Mathematical Society (1964; Zbl 0178.42902)]) and Ford’s construction (see L. R. Ford in [Automorphic functions, McGraw-Hill, New York (1929; JFM 55.0810.04)]). Each of these two methods yield a hyperbolically convex fundamental domain for any discrete subgroup of \(\text{SL}(2,\mathbb R)\). 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 \(\Gamma _{2}\) be a subgroup of \(\Gamma _{1}\) and \[ \Gamma_{1}=\Gamma_{2}\cdot \{L_{1},L_{2},\ldots,L_{m}\}. \] If \(\mathbb F\) is a fundamental domain of the bigger group \(\Gamma_1\), then the set \[ \mathcal{R}_{\Gamma}=\Biggl(\overline{\bigcup_{k=1}^{m}L_{k}(\mathbb{F})}\,\Biggr)^{o} \tag{(1)} \] is a fundamental domain of \(\Gamma_{2}\). One can ask at this juncture, is it possible to choose the right cosets suitably so that the set \(\mathcal R_{\Gamma }\) is hyperbolically convex? We will answer this question affirmatively for \[ \Gamma_{1}=\Gamma(1)\quad \text{and}\quad \mathbb{F}=\biggl\{\tau \in \mathbb{H}:|\tau|>1\,\&\, |\text{Re}(\tau)|<\frac{1}{2}\biggr\}. \]
MSC:
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
11F12 Automorphic forms, one variable
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30-04 Software, source code, etc. for problems pertaining to functions of a complex variable
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References:
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