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Understanding groups like \(\Gamma_ 0(N)\). (English) Zbl 0860.11019

Arasu, K. T. (ed.) et al., Groups, difference sets, and the Monster. Proceedings of a special research quarter, Columbus, OH, USA, Spring 1993. Berlin: Walter de Gruyter. Ohio State Univ. Math. Res. Inst. Publ. 4, 327-343 (1996).
This paper provides a completely trivial way of looking at the congruence groups \(\Gamma_0(N)\) of level \(N\) and their supergroups. Working with projective classes of lattices and their bases, the author describes \(\Gamma_0(N)\) as the group fixing \(L_1= \langle e_1,e_2 \rangle\) and \(L_N= \langle Ne_1,e_2 \rangle\). The number, \(N\), is the hyperdistance of this pair of lattices, since its logarithm is a metric. Lattices commensurable with \(L_1\) can be made into a graph by joining each pair of them by a line labelled with their hyperdistance. By restricting to prime hyperdistances, we obtain several graphs which, together, determine the resulting graph, the “big picture”. For prime \(p\), the resulting graph is a \(p+1\)-valent tree. This yields a nice description of the Hecke operator, \(T_n\), the Atkin-Lehner involutions, the lattices fixed by \(\Gamma_1(N)\) and \(\Gamma_0(N)\), and the \(PSL_2(\mathbb{R})\) normalizer of \(\Gamma_0(N)\). Finally Helling’s theorem [H. Helling, J. Lond. Math. Soc., II. Ser. 2, 67-72 (1970; Zbl 0189.09902)] – that the maximal discrete subgroups of \(PSL_2 (\mathbb{C})\), commensurable with the modular group, are just the conjugates of \(\Gamma_0 (N)+\) for square-free \(N\) – is deduced.
For the entire collection see [Zbl 0836.00028].
Reviewer: J.McKay (Montreal)

MSC:

11F06 Structure of modular groups and generalizations; arithmetic groups
11H06 Lattices and convex bodies (number-theoretic aspects)

Citations:

Zbl 0189.09902
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