Topological Symmetry Groups of Graphs in 3-Manifolds

TL;DR

This paper explores the possible topological symmetry groups of graphs embedded in 3-manifolds, establishing limitations and constructions for various groups within these spaces.

Contribution

It proves that certain groups cannot be realized as symmetry groups in any 3-manifold and constructs embeddings realizing arbitrary finite groups in hyperbolic rational homology spheres.

Findings

Alternating groups A_n are not realizable as topological symmetry groups in any 3-manifold.

Any finite group G can be realized as the symmetry group of some graph in a hyperbolic rational homology 3-sphere.

Abstract

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 {\Gamma} of some graph in a hyperbolic rational homology 3-sphere such that the topological symmetry group of {\Gamma} is isomorphic to G.