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How can I see that there is no lattice in $G=\mathrm{PSL}_2( \mathbb{R})$ which contains $\Gamma_1=\mathrm{PSL}_2( \mathbb{Z})$ properly, or equivalently, that $X_1 =\mathrm{PSL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is not a finite sheated nontrivial cover of a Riemann surface.

As a motivation, the eigenvalues of the Laplace Beltrami operator on $\mathrm{PSL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is conjectured to have only eigenvalues of multiplicity one. So now, if there exists $\Gamma \supset \Gamma_1$, an eigenvalue on $X_1$ can be associated to an irreducibel representation $\pi$ of $\mathrm{Ind}_{\Gamma_1}^{\Gamma} $, since $Ind_{\Gamma}^{G}$ $ \mathrm{Ind}_{\Gamma_1}^{\Gamma} 1 $ and $\mathrm{Ind}_{G}^{\Gamma}1 $ are isomorphic, and the eigenvalues appear with multiplicty being square of the dimension of $\pi$ (eigenfunctions =matrix coefficients).

Hence, if there would exists such a thing with a higher dimensional representation, the conjecture would be wrong. That's why the intuition that something should be known here.

What about $G$ reductive (see comment of David Loeffler)!

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I remember setting this (or, rather, the same question with SL(2) in place of PSL(2)) as an exercise to an undergraduate modular forms class once. Nobody solved it :-) If I remember correctly, the argument goes: first convince yourself that any such lattice must be contained in PSL(2, Q); then use the fact that PSL(2, Zp) is maximal compact in PSL(2, Qp) for every prime p. –  David Loeffler Apr 5 '11 at 7:25
    
So I guess, it is reasonable for $G$ reductive!? –  Marc Palm Apr 5 '11 at 7:38
    
@DavidLoeffler It's quite surprising that Modular forms are taught at the undergraduate level. Wouldn't it be too hard on students? –  S.C. Aug 24 '13 at 11:09

3 Answers 3

up vote 12 down vote accepted

We can say something stronger.

Theorem: (Helling 1976) Consider the family of subgroups of $SL_2(\mathbb{C})$ that are commensurable with a conjugate of $SL_2(\mathbb{Z})$. The maximal elements of this family under inclusion are precisely the conjugates of $\Gamma_0(N)^+$ for $N$ a squarefree integer.

Conway has a nice sketch of a proof in his paper "Understanding groups like $\Gamma_0(N)$" in the book Groups, difference sets, and the Monster. You can show using lattices that the commensurable groups act on the product of $p$-adic Bruhat-Tits buildings for primes $p$, and maximal groups are the groups that stabilize cells (i.e., finite products of edges).

I imagine you can do a similar calculation for higher-rank reductive groups, but the analysis may be harder, since the buildings are no longer trees.

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@Scott: Helling's work originated in his 1965 thesis and the paper based on it: Bestimmung der Kommensurabilit¨atsklasse der Hilbertschen Modulgruppe. (German), Math. Z. 92 (1966), 269–280. Some other references are provided in my answer, but like Helling's paper much of the earliest work was in German. –  Jim Humphreys Apr 5 '12 at 17:47

The existing answers all seem like huge overkill. The area of of the modular orbifold (quotient of the hyperbolic plane by $PSL(2, \mathbb{Z})$ is $\pi/3,$ so if it covers something, it should be an orbifold whose area divides that, so it should be a quotient by a triangle group, but it is easy to see that this is not possible by simple arithmetic.

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In fact you know that the purported quotient has one cusp, at least one cone point of order $2k$, and at least one of order $3m$, right? This must make the arithmetic almost trivial. –  Tom Church Apr 5 '12 at 19:41
    
@Tom: Yes, that's exactly right... –  Igor Rivin Apr 5 '12 at 20:46

The question in the header is interesting, but Scott's answer is directed at a somewhat different arithmetic question (also interesting). For your groups $G$ and $\Gamma_1$, or the similarly behaved groups $\rm{SL}_2(\mathbb{R})$ and $\rm{SL}_2(\mathbb{Z})$, the desired maximality of $\Gamma_1$ among lattices in $G$ is apparently due to Hecke in the late 1930s. But this kind of maximality question became embedded in much broader questions about lattices in Lie groups later on. The list of contributors is long and impressive, but it's useful for instance to look at a paper by Borel, Density and maximality of arithmetic subgroups, J. Reine Angew. Math. 224 (1966), 78–89. See especially Section 6 and Theorem 7 with its treatment of split groups.

Baily's Chicago student Nelo D. Allan wrote a thesis in this direction, with a useful overview in the Boulder volume: The problem of the maximality of arithmetic groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo. (1965), pp. 104–109, Amer. Math. Soc., Providence, R.I.

As these (and many related) references suggest, the maximality problem has often been studied in the context of semisimple algebraic groups over global and local fields, even though some of the results focus on a connected real semisimple Lie group and its discrete subgroups. In general, there are many lattices: discrete subgroups for which the quotient has finite measure (induced by Haar measure on the Lie group), either co-compact or not. Among the lattices are the arithmetic subgroups such as $\rm{SL}_2(\mathbb{Z})$, while arbitrary arithmetic groups are sometimes congruence subgroups and sometimes not. The rank one example considered here is actually more complicated than higher rank groups tend to be, leading to an extensive literature in modern times. But work like that of Borel and Serre (and many others from the 1960s onward) does provide a fairly unified treatment of the maximality question for arithmetic groups in all ranks, which is simplest to describe for split (Chevalley-type) groups.

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