A Bers type classification of big mapping classes

We show that the space of marked hyperbolic structures with the same ending geometry on an infinite type surface  is  locally path connected and connected  with respect to the Fenchel-Nielsen topology.  In particular, the space of   marked complete structures is locally path connected, connected. Since having the same ending geometry is a quasiconformal (qc) invariant, the space of such structures decomposes into the disjoint union of Teichmüller subspaces. 

We study the action of the mapping class group on these Teichmüller subspaces and, in particular, show that there are classes that are never qc.