The isometry group of the simply-connected Ricci-flat closed manifold

Question

If $M$ is a simply-connected Ricci-flat closed manifold, then is $I(M)$ the isometry group finite?

Answer 1

It is a theorem of Bochner that if $M$ is a compact Ricci-flat manifold, the dimension of the space of Killing fields of $M$ is $b_1(M)$. See Theorem 1.84 of Besse's "Einstein manifolds".

Since your $M$ is simply-connected (you just need $b_1(M)=0$), $M$ has no nonzero Killing fields. Therefore, $M$ has zero-dimensional group of isometries. Since the isometry group of a compact Riemannian manifold is a compact Lie group, $M$ has finite group of isometries.