Weyl law for SL(2,C) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:58:56Z http://mathoverflow.net/feeds/question/103074 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103074/weyl-law-for-sl2-c Weyl law for SL(2,C) Marc Palm 2012-07-25T06:39:44Z 2012-07-25T14:46:43Z <p>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.</p> <p>I have checked Mathscinet and couldn't find anything except for the main term.</p> http://mathoverflow.net/questions/103074/weyl-law-for-sl2-c/103106#103106 Answer by BR for Weyl law for SL(2,C) BR 2012-07-25T14:46:43Z 2012-07-25T14:46:43Z <p>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)\$.</p> <p>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 <a href="http://www.math.uni-bonn.de/people/mueller/papers/weyllaw.pdf" rel="nofollow">this survey</a> by M&uuml;ller (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.</p> <p>The book <a href="http://www.springer.com/mathematics/analysis/book/978-3-540-62745-6?changeHeader" rel="nofollow">Groups Acting on Hyperbolic Space</a> seems to (using <a href="http://books.google.com/books/about/Groups_Acting_on_Hyperbolic_Space.html?id=rYz0X9TP9cwC" rel="nofollow">Google Preview</a>) 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). </p>