Title:
The first part of this talk will be a quick introduction to the geometry and topology of big surfaces (surfaces with an infinitely generated fundamental group). Big surfaces include the Cantor tree, Loch Ness monster, and flute surfaces. Such surfaces admit a rich variety of hyperbolic geometric structures which are studied in a number of different ways.
The second part of this talk will be on big mapping class groups. The mapping class group of a surface Σ is its group of self-homeomorphisms modulo isotopy. If the fundamental group of Σ is finitely generated, the mapping class group is finitely generated and acts faithfully on the Teichmüller space of hyperbolic structures on Σ. On the other hand, for a big surface the mapping class group is infinitely generated (hence the name, big mapping class group) and does not act naturally on any Teichmüller space.
The final part of this talk is on our classification of big mapping classes and a construction of various spaces of hyperbolic structures for which the big mapping class group acts faithfully. As an application of our work, we show that a big mapping class group is not algebraically isomorphic to the modular group (quasiconformal mapping class group) of any hyperbolic surface. The final part is joint work with Yassin Chandran.
