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There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ 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 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 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 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 states that there are at most four exceptions with $\chi \geq -127$.

There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ 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 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 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 states that there are at most four exceptions with $\chi \geq -127$.

There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ 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 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 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 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 states that there are at most four exceptions with $\chi \geq -127$.

There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ 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 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 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 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 states that there are at most four exceptions with $\chi \geq -127$.

There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ 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 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 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 by Mostow rigidity).

There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ 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 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 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 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 states that there are at most four exceptions with $\chi \geq -127$.