Hyperbolic groups whose boundary is a Sierpinski -space
Joint work with Bena Tshishiku.
1. Sierpinski -space
Start with an -dimensional sphere. Remove a dense family of balls with disjoint interiors. Get . Up to homeo, balls need not be round. One merely needs that their diameters tend to 0.
Any homeo of permutes the distinguished peripheral spheres.
- Free groups have ideal boundary .
- If is a compact negatively curved -manifold with nonempty totally geodesic boundary, then .
- Let be a nonuniform lattice of isometries of . Then is cocompact on the complement of a union of horospheres, hence .
1.1. Cannon conjecture
What properties of the group follow from specifying the topology of the boundary ? This is what Cannon’s conjecture is about: if , must be a cocompact lattice in ?
Here is a topological variant of Cannonc’s conjecture.
Theorem 1 (Bartels-Lueck-Weinberger) If is torsion-free hyperbolic, and , and , then there exists a unique closed aspherical -manifold with .
Theorem 2 If is torsion-free hyperbolic, and , and , then there exists a unique aspherical -manifold with nonempty boundary with . Moreover, every boundary component of corresponds to a quasi-convex subgroup of .
3.1. Step 1
Kapovitch-Kleiner: is a relative -group, relative to the collection of stablizers of peripheral spheres.
3.2. Step 2
Realize as , where is a finite relative complex, relative to a finite subcomplex . We use the Rips complex for but the Bartles-Lueck-Weinberger complexes for parabolic subgroups .
3.3. Surgery theory
Browder-Novikov-Sullivan-Wall surgery theory provides obstructions to finding a manifold homotopy equivalent to . They belong to the space that appears in the algebraic surgery exact sequence
A similar exact sequence appears in 4-periodic surgery exact sequence, with replaced with a very similar (and with ). They have the same homotopy groups and differ only in their 0-spaces
There is a long exact sequence
It turns out that . Furthermore, thanks to the (L-theoretic) Farrell-Jones isomorphism conjecture (which holds for hyperbolic groups, Bartels-Lueck-Reich), . Hence , the obstruction vanishes, so there exists a homology manifold model for .
Bartels-Lueck-Reich cover groups. In the relative case (replace spheres with Sierpinski spaces), much of the argument carries over, but the first step.