Notes of Grigori Avramidi’s Cambridge lecture 23-06-2017

Topology of ends of nonpositively curved manifolds

Joint work with T. Nguyen Pham.

I am interested in complete Riemannian manifolds with curvature in {[-1,0]}, and finite volume.

Example. Product of two hyperbolic surfaces. The end is homeomorphic to {N\times[0,+\infty)}, with some extra structure: {N} is made of two pieces.

More generally, for locally symmetric spaces of noncompact type, lifts of ends are homeomorphic to {N\times[0,+\infty)}, with {N} a wedge of spheres. This description goes back to Borel-Serre.

1. Thick-thin decomposition

Gromov-Schroeder: assume there are no arbitrarily small geodesic loops. Then the thin part is homeomorphic to {N\times[0,+\infty)}, with {N} a closed manifold.

The condition is necessary. Gromov gives an example of a nonpositively curved infinite type graph manifold of finite volume.

Theorem 1 (Avramidi-Nguyen Pham) Under the same assumptions, any map of a polyhedron to the thin part of the universal cover {\tilde M} can be homotoped within the thin part into a map to an {\lfloor \frac{n}{2}\rfloor}-dimensional complex, {n=dim(M)}.

Consequences:

  1. If {n\leq 5}, each component of the thin part is aspherical and has locally free fundamental group.
  2. {H^k(B\Gamma,{\mathbb Z} \Gamma)=0} for all {k<\frac{n}{2}}.
  3. {dim(B\Gamma)\geq \frac{n}{2}}.

2. Proof

Maximizing the angle under which two visual boundary points are seen gives Tits distance, and the corresponding path metric {Td}.

In the universal cover, the thin part is the set of points moved less than {\epsilon} away by some deck transformation {\Gamma}. Isometries are either hyperbolic (minimal displacement is achieved) or parabolic (infimal displacement is 0). Parabolic isometries have a nonempty fix-point set at infinity. At each point {x}, the subgroup generated by isometries moving {x} no more than {\epsilon} is virtually nilpotent, hence virtually has a common fixed point at infinity. This allows to define a discontinuous projection to infinity. The point is to show that the image has dimension {<\lfloor \frac{n}{2}\rfloor}.

2.1. Busemann simplices

If {h_0} and {h_1} are Busmeann functions, {t_0h_0+t_1h_1} need not be a Busemann function again, but on each sphere, there is a unique point where it achieves its minimum, and tis point depends in a Lipschitz manner on {t_0,t_1}. This defines an arc in Tits boundary, hence simplices {\sigma}. We claim that

\displaystyle  \begin{array}{rcl}  hom-dim(Stab(\sigma))+dim(image(\sigma))\leq n-1. \end{array}

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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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