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The maximal discrete extension of \(\mathrm{SL}_2(\mathcal {O}_K)\) for an imaginary quadratic number field \(K\). (English) Zbl 1457.11029

Let \(K\) be an imaginary quadratic number field and let \(\mathcal{O}_K\) be its ring of integers. Let \(\Gamma_K\) denote the group \(\mathrm{SL}_2(\mathcal{O}_K)\) and let \(\Gamma^*_K\) be its maximal discrete extension inside \(\mathrm{SL}_2(\mathbb{C})\). A description of \(\Gamma_K^*\) is provided by [J. Elstrodt et al., Groups acting on hyperbolic space. Harmonic analysis and number theory. Berlin: Springer (1998; Zbl 0888.11001)]. (See Proposition 4.2, p.328.) In this paper the authors provide an alternative description using matrices which can be be viewed as a generalization of Atkin-Lehner involutions. It turns out that \(\Gamma_K\) is a normal subgroup of \(\Gamma_K^*\) and that the quotient group is isomorphic to a subgroup of the elementary \(2\)-subgroup of the ideal class group of \(\mathcal{O}_K\). More precisely \[ \Gamma_K^*/\Gamma_K \cong C_2^{\nu},\] where \(\nu\) is determined by \(d_K\), the discriminant of \(K\). No maximal discrete extension of \(\Gamma_K\) exists inside \(\mathrm{GL}_2(\mathbb{C})\). The authors use their approach to provide a natural characterization of the groups \(\Gamma_K,\; \Gamma_K^*\) in the special orthogonal group \(\mathrm{SO}(1,3)\).

MSC:

11F06 Structure of modular groups and generalizations; arithmetic groups

Citations:

Zbl 0888.11001
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References:

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