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.



Weyl law for SL(2,C)
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](http://www.math.uni-bonn.de/people/mueller/papers/weyllaw.pdf) 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](http://www.springer.com/mathematics/analysis/book/978-3-540-62745-6?changeHeader) seems to (using [Google Preview](http://books.google.com/books/about/Groups_Acting_on_Hyperbolic_Space.html?id=rYz0X9TP9cwC)) 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). 


