Fake Congruence Subgroups and the Hurwitz Monodromy Group
Vol. 6 (1999), No. 3, Page 559--574.
Berger, Gabriel
Fake Congruence Subgroups and the Hurwitz Monodromy Group
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Abstract:
Suppose G is a finite group, embedded as a transitive subgroup of Sn for some n. Suppose in addition that (C1,…,C4) is a quadruple of conjugacy classes of G. In earlier papers ([F], [D-F], [B-F]), it was shown that to these data one can canonically associate a finite index subgroup of PSL2(Z). For example, when N is an odd integer, G is the dihedral group DN and the conjugacy classes all consist of involutions, the associated subgroup is Î_0(N). In this paper we investigate the case in which G is the semidirect product of the abelian group \mathbb{Z}[ζ_d]/\mathcal{N} (where ζ_d is a primitive d'th root of unity and \mathcal{N} is an ideal of \mathbb{Z}[ζ_d] relatively prime to d) and the cyclic group \langle ζ_d \rangle. We relate the corresponding subgroup of PSL_2(\mathbb{Z}) to the "fake congruence subgroups" described in \cite{B2}. Specifically, if we let \mathcal{C} denote the conjugacy class of ζ_d in the multiplicative subgroup \langle ζ_d \rangle and choose our conjugacy classes to be (\mathcal{C}, \mathcal{C}, \mathcal{C}, \mathcal{C}^{-3}) , then the subgroup is in fact Î_0(\mathcal{N}) (defined originally in \cite{B2}; see section 2).
Keywords: Burau representation, fake congruence subgroup, Hurwitz space, modular tower
Mathematics Subject Classification (1991): Primary 11F06; Secondary 20F36
Mathematical Reviews Number: MR1726683
Received: 1998-08-08
