Farb-Weinberger's quantitative symmetry-gap conjecture

Let $M$ be a smooth closed, aspherical, smoothly irreducible manifold such that $pi_1(M)$ contains no nontrivial normal abelian subgroups. Then there exists $C>1$ depending only on $pi_1(M)$ such that if $g$ is any Riemannian metric on $M$ with

$$[\operatorname{Isom}(\widetilde{M}):\pi_1(M)]\geq C,$$

then $M$ is isometric to a locally symmetric space of noncompact type. Farb-Weinberger's conjecture. The quantitative symmetry gap asserts that sufficiently large isometry groups of universal covers force a closed, aspherical, smoothly irreducible manifold to be locally symmetric. It refines the qualitative characterization of locally symmetric spaces and is presented here as a conjecture.