** The borders of Outer Space **

Joint work with Kai-Uwe Bux and Peter Smillie.

**1. Duality groups **

I am interested in Poincare duality. For a group, assume is a smooth -manifold, then

Bieri-Eckmann observed that is suffices that acts freely cocompactly on a contractible space whose compactly supported cohomology vanishes in all degrees but , and is torsion free. Then is a duality group.

If the action is merely proper and cocompact, is a virtual duality group. Borel-Serre used this for lattices. Bestvina-Feighn used this to show that $latex {Out(F_n) is a virtual duality group. Mapping class groups also act on a contractible space.

To achieve cocompactness, Borel-Serre added to the symmetric space copies of Euclidean space forming a rational Euclidean building. The resulting bordification of the quotient is a manifold, with boundary homotopy equivalent to a wedge of spheres (Solomon-Tits). Instead, Grayson constructed an invariant cocompact subset of symmetric space.

Grayson’s work was used by Bartels-Lueck-Reich-Ruping to prove Farrel-Jones for }&fg=000000$Sl(n,{\mathbb Z})$latex {.

For }&fg=000000$Out(F_n)J_n\subset X_n$latex {. It is much easier and gives more information on the boundary.

\section{Construction}

}&fg=000000$X_nJ_n$ is obtained by chopping off some of their corners.

A core graph is a subgraph such that, when one shrinks it, one gets out of Outer Space. Only corner facing core graphs need be chopped off.

The boundary appears as a union of contractible walls (every intersection of walls is contractible).

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