Continuous and discontinuous functions on deformation spaces of Kleinian groups

Abstract

To understand the topologies of deformation spaces of Kleinian groups, it is helpful to make use of functions defined on deformation spaces and to study their continuity and discontinuity. In this chapter, we focus on the three functions, the length function, the end invariant function, and the two-variable Cannon–Thurston map, restricting our attention to deformation spaces of Kleinian surface groups, and study how they behave on the boundaries of deformation spaces. We show that although the length function is continuous on the entire deformation space, both the end invariant function and the two-variable Cannon–Thurston map have discontinuous points, reflecting the difference of geometric limits and algebraic limits. Furthermore, we show, using several illuminating examples, that the end invariant function and the two-variable Cannon–Thurston map have different sensibilities towards geometric limits.