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Are there any estimates for the eigenvalues of the Laplace operator for $\Gamma \backslash SL(2, \mathbb{C})/SU(2)$ known beyond the main term? Here, $\Gamma$ should be congruence subgroup in $SL(2,o)$ for $o$ the ring of integers in an imaginary quadratic field.

I have checked Mathscinet and couldn't find anything except for the main term.

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up vote 4 down vote accepted

For general compact manifolds (of dimension $n$), the error term (on the number of eigenvalues less than $T^2$, counted with multiplicity) is $O(T^{n-1})$. So for $\Gamma$ co-compact, the error term is $O(T^2)$.

For merely co-finite $\Gamma$, it should be possible to bound the error term, but I am unaware of any results. In Section 4 of this survey by Müller (by the way, equation 1.3 is Weyl's Law for compact manifolds with the error term, and Section 2 sketches a different approach for compact locally symmetric spaces), he sketches a proof of a strong form of the law (using the Selberg Zeta function) and implies that the argument would work for other rank-one groups.

The book Groups Acting on Hyperbolic Space seems to (using Google Preview) prove the strong form of the law in the co-compact case (Section 5.5), but only seems to prove the weak form in the co-finite case (Section 8.9).

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... and $T^{n-1}$ is essentially optimal right, perhaps rather $T^{n-1}/log T$? I was aware of all this, but thanks already. – Marc Palm Jul 25 '12 at 16:00
Yeah, $O(T^{n-1})$ is optimal for compact spaces because of spheres, but I think conjecturally tighter bounds exist for certain hyperbolic spaces (and you are correct that for $SL_2(\mathbb Z)\backslash \mathfrak h$, Selberg obtained $O(T/{\rm log}\ T)$, so something better is certainly possible). I think it might be something like GRH (though more tractable), where getting nonvanishing of an $L$-function at the edge of the strip is hard, and making any improvement on that virtually impossible, even though we expect much more. – B R Jul 25 '12 at 17:14

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