A Bers type classification of big mapping classes

Abstract: For an infinite type surface $\Sigma$, we consider the space of (marked) convex hyperbolic structures on $\Sigma$, denoted $H(\Sigma)$, with the Fenchel-Nielsen topology. The (big) mapping class group acts faithfully on this space allowing us to investigate a number of mapping class group invariant subspaces of $H(\Sigma)$ which arise from various geometric properties (e.g. geodesic or metric completeness, ergodicity of the geodesic flow, lower systole bound, discrete length spectrum) of the hyperbolic structure. In particular, we show that the space of geodesically complete convex hyperbolic structures in $H(\Sigma)$ is locally path connected, connected and decomposes naturally into Teichm\"uller subspaces. The big mapping class group of $\Sigma$ acts faithfully on this space allowing us to classify mapping classes into three types ({\it always quasiconformal, sometimes quasiconformal, and never quasiconformal}) in terms of their dynamics on the Teichm\"uller subspaces. Moreover, each type contains infinitely many mapping classes, and the type is relative to the underlying subspace of $H(\Sigma)$ that is being considered. As an application of our work, we show that if the mapping class group of a general topological surface $\Sigma$ is algebraically isomorphic to the modular group of a Riemann surface $X$, then $\Sigma$ is of finite topological type and $X$ is homeomorphic to it. Moreover, a big mapping class group can not act on any Teichm\"uller space with orbits equivalent to modular group orbits.