TITLE: Multiplicity of Laplace eigenvalues
QUESTION [9 upvotes]: Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher dimensional manifolds) whose spectrum I have seen evaluated explicitly have highly repeated eigenvalues. They also happen to be quite symmetric (non-trivial isometry group), which was instrumental in the calculation of the spectrum in the first place. I have heard many people quote this heuristic: the more symmetric a manifold is, the higher the chances of spectral multiplicity. I was wondering 

Is there a way to make this heuristic precise?
What about the converse? If a manifold has high spectral multiplicity, does it need to have a nontrivial isometry group?
If the answer to 2. is negative, and there are large classes of examples of surfaces/manifolds that are not symmetric at all, but have high spectral multiplicity, is there some other property shared by these examples which is causing the multiplicity? In other words, are there other criteria apart from symmetry that would make one suspect spectral multiplicity?  

Edit: Okay, I just found High multiplicity eigenvalue implies symmetry? So, I know that the answer to 2. is negative. Also, Liviu Nicolaescu answers 1. below. I am still confused about 3. In other words, is there any criterion other than symmetry, that, if you know that a metric satisfies, you would suspect high spectral multiplicity?

REPLY [9 votes]: The group of  isometries of a compact Riemann manifold is a compact Lie group.  This group acts on  the eigenspaces of the Laplacian. If the group  of isometries   is non-abelian, then it is natural to expect that that some eigenspaces will have large dimensions since  most of the nontrivial irreducible representations of  a compact  non-abelian Lie group have  dimensions $\geq 2$.
Looking for Laplacians with multiple eigenvalues is a bit like looking for a needle in a haystack since  a result of K. Uhlenbeck states  that for a generic Riemann metric on a given compact manifold the spectrum of the Laplacian  will not have multiple eigenvalues.