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Congruence subgroups associated to the Monster. (English) Zbl 1099.11021

Let \(\Delta\) denote the set of all genus zero subgroups \(G\) of \(\Lambda=\text{PSL}_2(\mathbb{R})\) for which \[ \Gamma_0(m) \leq G \leq N(\Gamma_0(m)), \] for some positive integer \(m\), where \(N(\Gamma_0(m))\) is the normalizer of \(\Gamma_0(m)\) in \(\Lambda\). (The genus of a (discrete) subgroup \(S\) of \(\Lambda\) is the genus of the Riemann surface \(\mathbb{H}^2/S\), where \(\mathbb{H}^2\) is the hyperbolic \(2\)-space.)
In this paper the authors determine all such \(m\) and, for each such \(m\), they list all the members of \(\Delta\). In the process they list also all those members of \(\Delta\) of width \(1\) at \(\infty\). The proofs involve computations using GAP.
Their results answer an earlier question of Conway and Norton from “moonshine theory”.
The authors discuss the overlap between their results and those of C. J. Cummins in “Congruence subgroups of groups commensurable with \(\text{PSL}_2(\mathbb{Z})\) of genus \(0\) and \(1\)” [Exp. Math. 13, No. 3, 361–382 (2004; Zbl 1099.11022)].

MSC:

11F06 Structure of modular groups and generalizations; arithmetic groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
11F22 Relationship to Lie algebras and finite simple groups

Citations:

Zbl 1099.11022
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References:

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