Does every triangulable manifold have a vertex-transitive triangulation?


When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically as a simplicial complex such that the automorphism group of the simplicial complex is transitive on the vertices or 0-simplexes of the complex.
This could also be formulated for other categories, like the category of simplicial sets, or the category of maps on surfaces (once you define G-actions suitably) but we'll stick with graphs and simplicial complexes here.


Two of the reasons I was interested in this problem is the work that's been done on local computability of characteristic classes of manifolds by Gelfand and others, as well as the conjecture of Kahn, Saks and Sturtevant on vertex transitive nonevasive complexes.


I'm surprised no one has really raised the question in print before, to my knowledge. Once, I thought I had an idea for a proof that some 2-manifolds can't be suitably triangulated, by focusing on surfaces which can't be realized suitably by regular maps (or flag transitive triangulations) and showing some of these surfaces can't be vertex transitively triangulated in any other way. I hope to return to this and complete the details sometime soon.



Does every triangulable manifold have a vertex-transitive triangulation?
There [exists](https://mathscinet.ams.org/mathscinet-getitem?mr=953960) many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence [trivial mapping class group](https://www.ams.org/journals/jams/1997-10-01/S0894-0347-97-00206-3/). $M$ cannot be homeomorphic to a simplicial complex $\tau$ which admits a vertex-transitive automorphism group $G$ (which must be non-trivial). Then the quotient $M/G = \tau/G$ is a 3-orbifold which by the [orbifold theorem](https://en.wikipedia.org/wiki/Orbifold#3-dimensional_orbifolds) must be hyperbolic. But this implies that the symmetry group was conjugate in the mapping class group to a group of isometries, a contradiction.


I imagine that this ought to be true for [symmetry-free hyperbolic manifolds](https://mathscinet.ams.org/mathscinet-getitem?mr=2132171) in any dimension >2, but I’m not quite sure how to prove that the automorphism group of the triangulation induces non-trivial outer automorphisms of the fundamental group (equivalently non-trivial isometries up to homotopy by Mostow rigidity).


For the 2-dimensional case, it does appear to be open in general which closed surfaces admit a vertex-transitive triangulation. [This paper](https://arxiv.org/abs/1001.2777) states that there are at most four exceptions with $\chi \geq -127$.


