Some problems in hyperbolic 3-manifolds

Geometry from topology

1. A "gluing problem": Let M₁,M₂ be compact 3-manifolds and f:¶M₁ ® ¶M₂ be a homeomorphism. Give a uniform bilipschitz model for M = M₁ È_f M₂, with estimates depending only on the topology of M_i and the map f.
2. Incompressible case: If ¶M_i are incompressible, then the gluing problem reduces to (a) the model manifold constructed by Brock-Canary-Minsky for the solution of the Ending Lamination Conjecture, and (b) Thurston's "bounded image theorem" which says when ¶M_i is acylindrical that the image of the "skinning map" in Teich(¶M_i) is compact. Given a topological description of M_i, can one give a quantitative bound on the diameter of the skinning map image?

Structure of deformation spaces

1. Give a complete topological description of this space. (This is perhaps impossible. Although the space is enumerated by end-invariants, its topological structure is quite complicated. The end-invariants vary discontinuously, and Bromberg has shown the deformation space is not locally connected).
2. Bers slice: If M=S×[-1,1] then quasifuchsian structures are parametrized by Teich(S)×Teich(S). A Bers slice B_X is the set parametrized by {X}×Teich(S). It is homeomorphic to \mathbb R^6g(S)-6. Is the closure of B_X a ball?
3. What is the Hausdorff dimension of the boundary of the deformation space?
4. Let MCG(M) denote the group of homeomorphisms of M modulo homotopy. Describe the dynamics of MCG(M) on the representation space. Do the convex-cocompact representations comprise the largest open set on which the action is properly discontinuous?

Representations of surface groups

1. Cannon-Thurston maps: If r is discrete and faithful, show that there is a continuous equivariant map from S¹ to S² (where p₁(S) acts on S¹ via any Fuchsian representation and PSL(2,\mathbb C) acts on S² by Möbius transformations). This is known when r has "bounded geometry" (lower bound on injectivity radius), and there have been a number of extensions, notably a complete solution by McMullen when S is a one-holed torus. There is a general form of this conjecture for any finitely-generated group replacing S¹ by the Gromov boundary. A positive solution would complete our picture of the action of a Kleinian group on the 2-sphere.
2. Bowditch's conjecture: Consider the quantity
   

   rr = inf g  lr(g) l0(g)

   where g varies over simple (non-self-intersecting) closed curves in S, l_r denotes translation distance of the image in \mathbb H³, and l₀ denotes translation length in some Fuchsian structure on S. Show that r_r > 0 if and only if r is quasi-fuchsian. This is known if r is discrete and faithful, but not for general representations. I.e. show that if r is indiscrete or nonfaithful then r_r=0. This would be trivial if g were not required to be simple.
3. Simple loop conjecture for representations: philosophically related to Bowditch's conjecture. If the kernel of r is nonempty, does it contain a simple element?
4. An interesting special case of (*): Consider the action of MCG(one-holed torus)=PSL(2,\mathbb Z) on the representation space for the one-holed torus. This space may be parametrized (through traces of generators) as \mathbb C³, and the group action is generated by (x,y,z) ® (x,y,xy-z) and (x,y,z) ® (y,z,x).