Let $M$ be a compact manifold and $\text{diff}(M)$ its diffeomorphism group. Various quotients of $\text{diff}(M)$ appear in the literature, oftentimes with geometric significance. A well-known example is the space $\text{diff}(M)/\text{diff}_\mu(M)$, where $\mu$ is a measure on $M$ of total volume one and $\text{diff}_\mu(M)$ is the group of diffeomorphisms of $M$ that preserve $\mu$. (See Hamilton's Nash-Moser paper for more details.)

**Questions:**

(1) Is $\text{diff}(M)/\text{diff}_\mu(M)$ a tame Fréchet manifold?

(2) Is $\text{diff}_\mu(M)$ a tame Fréchet submanifold of $\text{diff}(M)$? (Hamilton's paper says that this is an open problem. Has any progress been made on this since 1982?)

(3) If $G$ is a subgroup of the diffeomorphism group, when is $\text{diff}(M)/G$ a tame Fréchet manifold?

I know only of a result that addresses (3) on totally different lines. This is the Gleason-Yamabe theorem. It says that if $G$ is locally compact, then for any open neighborhood $U$ of the identity there exists a subgroup $G'$ of $G$ contained in $U$ and a compact $G'$-normal subgroup $K\subset U$ such that $G'/K$ is isomorphic to a Lie group. (Terence Tao has written about this theorem in connection with Hilbert's Fifth Problem.) Since I want to quotient out the full diffeomorphism group by a subgroup, this won't do.