Minimal and non-minimal automorphism groups of homogeneous structures

A Hausdorff topological group G is said minimal if G does not admit any strictly coarser Hausdorff group topology.

Examples include the isometry group of the Urysohn sphere, due to Uspenskij, and Aut(M) for M stable and w-categorical, a deep fact due to Ben Yacov and Tsankov.

I will discuss joint work in preparation with with Zaniar Ghadernezhad where we explore the question of minimality of automorphism groups beyond stability and Roelcke precompactness using more modest methods. We generalize Uspenskij's result to the unbounded Urysohn space and give a sufficient condition for minimality based on the existence of a free independence relation in the sense of Conant.