$\DeclareMathOperator\vol{vol}\DeclareMathOperator\diam{diam}$I have a sequence of compact Riemannian manifolds $M_n$ with $\diam (M_n) \to 0$ and finite fundamental groups $\pi_1 (M_n)$ so that their universal covers $L_n$ are compact. Suppose $\diam ( L_n ) = 1$ for all $n$, and their volumes are normalized so that $\vol(L_n)=1$.
Is it true that the family of metric measure spaces $L_n$ satisfies some concentration of measure phenomenon? Are they a Levy family? This is closely related to this other question: Diameter of universal cover .
