Path-connectedness of Dehn filling parameter space

Determine whether the parameter space of hyperbolic Dehn fillings—defined as the subset of S^2 of slopes (p,q) for which the metric completion M_{p,q} of a cusped hyperbolic 3-manifold M supports a hyperbolic structure—is path-connected.

Background

In the discussion of hyperbolic Dehn filling, the paper distinguishes between the local Thurston slice near the ideal point and a broader "parameter space" comprising all slopes (p,q) for which the metric completion supports a hyperbolic structure. The author notes that while the Thurston slice is well-understood locally, the global properties of this parameter space are poorly understood.

The paper explicitly flags uncertainty about the topology of this parameter space, highlighting that even basic questions such as path-connectedness are unresolved. Clarifying the path-connectedness would advance understanding of the global structure of Dehn filling parameters that yield hyperbolic manifolds.