Timeline for High multiplicity eigenvalue implies symmetry?
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| Mar 18, 2012 at 3:32 | comment | added | Alex Eskin | @Igor: The most natural way to estimate the multiplicity of the eienvalues is to use the trace formula, which relates it to a sum over the closed geodesics (i.e. the length spectrum). But the problem is that you get a sum over exponentially many geodesics, and you end up trying to add up e^\lambda terms of size O(1), while the actual answer is also O(1). This is why it is so hard. | |
| Mar 18, 2012 at 2:54 | comment | added | Igor Rivin | Well, the funny thing is that a result of this sort DOES hold for the length spectrum: Takeuchi (I believe) showed that if a surface has linear number of traces smaller than $N,$ if and only if it is arithmetic, which is morally a closely related result. | |
| Mar 17, 2012 at 23:04 | history | answered | Alex Eskin | CC BY-SA 3.0 |