Georgia Institute of Technology College of Sciences

An infinite-type surface is a surface whose fundamental group is not finitely generated. These surfaces are “big,” having either infinite genus or infinitely many punctures. Recently, it was shown that mapping class groups of these infinite-type surfaces have a wealth of subgroups; for example, there are infinitely many surfaces whose mapping class group contains every countable group as a subgroup. By extending a theorem for finite-type surfaces to the infinite-type case — the Nielsen realization problem — we give topological obstructions to continuous embeddings of topological groups, with a few interesting examples.