Could someone give examples of non-Riemannian manifolds that are Alexandrov spaces with $\mathrm{sec}\geq-1$ and the first eigenvalue equal to $(n-1)^2/4$?

I'm not sure if I understand your question, but one might be able to take $\mathbb{H}^n$ modulo an appropriate finite group. It would help if you could give a reference for the definition of the eigenvalue for a non-Riemannian Alexandrov space.
– Ian AgolJan 15 '13 at 18:21

"sharp spectral gap and Li-Yau's estimate on Alexandrov spaces" ,the first page you can see the definition
– jiangsaiyinJan 16 '13 at 2:39