## 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

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