The Reconstruction and Realization of Topological Groups

A topological group is a group equipped with a topology so that the group operations are continuous. The symmetries (or automorphisms) of any given geometric object have a natural group structure, and often a canonical topology making them into a topological group. Specifically, the automorphism groups of a countable structure is a topological group with the pointwise convergence topology, and the autohomeomorphism group of a compact space is a topological group with the compact-open topology. Here we investigate when these canonical topologies are minimal or the minimum amongst all Hausdorff group topologies. Further, given an abstract topological group, we aim to realize it as the symmetry group of some geometric object with its canonical topology. We examine two such classes of objects: the automorphism groups of graphs with the pointwise convergence topology and the autohomeomorphism groups of continua (compact and connected spaces) with the compact-open topology.