Notes of Jean-François Lafont’s Cambridge lecture 22-06-2017

Hyperbolic groups whose boundary is a Sierpinski {n}-space

Joint work with Bena Tshishiku.

1. Sierpinski {n}-space

Start with an {(n+1)}-dimensional sphere. Remove a dense family of balls with disjoint interiors. Get {S_n}. Up to homeo, balls need not be round. One merely needs that their diameters tend to 0.

Any homeo of {S_n} permutes the distinguished peripheral spheres.

Examples.

  1. Free groups have ideal boundary {S_0}.
  2. If {M} is a compact negatively curved {n}-manifold with nonempty totally geodesic boundary, then {\partial\tilde M=S_{n-2}}.
  3. Let {\Gamma} be a nonuniform lattice of isometries of {H^n}. Then {\Gamma} is cocompact on the complement {A} of a union of horospheres, hence {\partial A=S_{n-2}}.

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 {\partial\Gamma=S^2}, must {\Gamma} be a cocompact lattice in {H^3}?

Here is a topological variant of Cannonc’s conjecture.

Theorem 1 (Bartels-Lueck-Weinberger) If {\Gamma} is torsion-free hyperbolic, and {\partial \Gamma=S^{n-1}}, and {n\geq 6}, then there exists a unique closed aspherical {n}-manifold {M} with {\Gamma=\pi_1(M)}.

2. Result

Theorem 2 If {\Gamma} is torsion-free hyperbolic, and {\partial \Gamma=S_{n-2}}, and {n\geq 7}, then there exists a unique aspherical {n}-manifold {M} with nonempty boundary with {\Gamma=\pi_1(M)}. Moreover, every boundary component of {M} corresponds to a quasi-convex subgroup of {\Gamma}.

3. Proof

3.1. Step 1

Kapovitch-Kleiner: {\Gamma} is a relative {PD(n)}-group, relative to the collection of stablizers of peripheral spheres.

3.2. Step 2

Realize {\Gamma} as {\pi_1(X)}, where {X} is a finite relative {PD} complex, relative to a finite subcomplex {Y\subset X}. We use the Rips complex for {B\Gamma} but the Bartles-Lueck-Weinberger complexes for parabolic subgroups {\Lambda_i}.

3.3. Surgery theory

Browder-Novikov-Sullivan-Wall surgery theory provides obstructions to finding a manifold homotopy equivalent to {X}. They belong to the space {S(X)} that appears in the algebraic surgery exact sequence

\displaystyle  \begin{array}{rcl}  \cdots\rightarrow H_n(X,L_\cdot)\rightarrow H_n({\mathbb Z}\Gamma)\rightarrow S(X)\rightarrow H_{n-1}(X,L_\cdot)\rightarrow\cdots. \end{array}

A similar exact sequence appears in 4-periodic surgery exact sequence, with {L} replaced with a very similar {\bar L} (and {S(X)} with {\bar S(X)}). They have the same homotopy groups and differ only in their 0-spaces

\displaystyle  \begin{array}{rcl}  \textrm{for }L_\cdot, ~G/TOP ; \quad \textrm{for }\bar L_\cdot, ~G/TOP\times L_0({\mathbb Z}). \end{array}

There is a long exact sequence

\displaystyle  \begin{array}{rcl}  \cdots\rightarrow H_n(X,L_0({\mathbb Z}))\rightarrow S_n(X)\rightarrow\bar S_n(X)\rightarrow H_{n-1}(X,L_0({\mathbb Z}))\rightarrow\cdots. \end{array}

It turns out that {H_n(X,L_0({\mathbb Z}))=H_n(X,{\mathbb Z})=0}. Furthermore, thanks to the (L-theoretic) Farrell-Jones isomorphism conjecture (which holds for hyperbolic groups, Bartels-Lueck-Reich), {\bar S_n(X)=0}. Hence {S_n(X)=0}, the obstruction vanishes, so there exists a homology manifold model for {B\Gamma}.

3.4. {CAT(0)} groups

Bartels-Lueck-Reich cover {CAT(0)} groups. In the relative case (replace spheres with Sierpinski spaces), much of the argument carries over, but the first step.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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