Special Relativity states that all things exist relative to one another and so no true fixed points can exist or at the very least be found.
I feel this can be contradicted when looking at space-time singularities. A space-time singularity being a point at which the gravitational pull becomes infinite (or pseudo-infinite), if this is true then wouldn't the fabric of space-time be fixed around that point, any movement of the singularity would then be directly expressed by the fabric of space-time and vice-versa. This would then be a "fixed-point", unfortunately this would only be useful if the fabric of space-time was unchangeable but for the theory of gravity to function this cannot be the case.
This is why I describe these theoretical points as sort of fixed, the fabric of space-time would be directly tied to a collection of matter but gravity still bends space-time. If this theory does make sense, it may lead to some useful revelations about how Universe Expansion works and why the expansion is accelerating.
I freely admit I have no given calculations for this, I feel this is a job for someone far smarter than me.