Sortable elements in infinite Coxeter groups.

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group $ W$ $ \pi_\downarrow^c$ to the unique maximal $ c$ $ \pi_\downarrow^c$-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of $ \pi_\downarrow^c$ [ABSTRACT FROM AUTHOR]