There is a well known problem of LeBrun-Salamon:
are there any non-symmetric compact quaternionic-Kahler
manifolds of positive scalar (and Ricci) curvature?
It is hard and still unsolved:
[Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups?](https://mathoverflow.net/questions/78793/quaternionic-kahler-metrics-whose-universal-covers-have-only-discrete-isometry-g)


The symmetric compact quaternionic-Kahler manifolds
("Wolf spaces") are understood and classified. However,
for each Wolf space there is the dual symmetric
space, say, $G/H$, which is quaternionic-Kahler
of negative scalar (and Ricci) curvature.


For any lattice $\Gamma\subset G$,
the double quotient $\Gamma\backslash G/H$
is a locally symmetric quaternionionic-Kahler
orbifold of negative curvature and finite volume.
However, it can have cusp points, and then it 
is non-compact.


Are there any compact locally symmetric
quaternionionic-Kahler orbifolds? Manifolds?
I could not find a reference to either existence
or non-existence results.



Compact quaternionic Kahler manifolds of negative curvature: examples
Any (Riemannian) symmetric space admits a cocompact lattice. This is due to A. Borel, [Compact Clifford-Klein forms of symmetric spaces,](http://www.sciencedirect.com/science/article/pii/0040938363900260) Topology 2, 1963, pp.111-122. The quaternionic hyperbolic space is symmetric and quaternionic-Kahler.


