In 1980s, Thurston's formulated the geometrization conjecture for 3-manifolds, and proved the hyperbolization theorem. The keys to Thurston's proof are two bounded results for certain deformation spaces of Kleinian groups. In early 1990s, motivated by Thurston's boundedness theorem and the Sullivan dictionary, McMullen conjectured that Sierpinski carpet hyperbolic components of rational maps are bounded.

In this talk, I will start with a historical discussion on a general strategy of the proof of Thurston's boundedness theorem. I will then explain some difficulties in the rational map setting to carry out the strategy, and discuss how to overcome this. In particular, I will explain how to use a variation of the uniform a priori Siegel bound to control the post-critical set of maps in any hyperbolic component of disjoint type, which will imply McMullen's conjecture in the disjoint type setting. This is based on a joint work with D. Dudko.