Topological Symmetry Groups of Graphs in 3-Manifolds

We prove that for every closed, connected, orientable, irreducible 3-manifold, there exists an alternating group A_n which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group G, there is an embedding Γ of some graph in a hyperbolic rational homology 3-sphere such that the topological symmetry group of Γ is isomorphic to G.

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