# mg.metric geometry – Do Compact Universal Covers have Concentration of Measure phenomenon?

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 .