Continuous and discontinuous functions on deformation spaces of Kleinian groups (Q2055092)

The paper under review is an interesting survey on the topologies of deformation spaces of Kleinian surface groups. It gives an original overview of spaces of Kleinian groups and functions defined on them, with a special emphasis on the continuity and discontinuity properties of these functions. The main idea conveyed is that in several settings of this kind, discontinuity phenomena are related to the fact that the spaces on which the functions are defined carry more than one natural topology. In the present case, these are the algebraic and geometric topologies. Among the topics that are discussed in detail in this paper are the properties of length functions on deformation spaces, end invariants as functions that can be used in the parameters for these spaces, and the deformation theory of the Cannon-Thurston maps. The author proves in particular that while the length functions are continuous on the entire deformation spaces, both the end invariant function and the two-variable Cannon-Thurston map have discontinuity points, and that this reflects the difference of geometric limits and algebraic limits. Furthermore, he shows, using several illuminating examples, that the end invariant function and the two-variable Cannon-Thurston map have different behaviors with respect to geometric limits. For the entire collection see [Zbl 1466.57001].