Notes of Karen Vogtmann’s second Cambridge lecture 23-06-2017

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 ${M=B\pi}$ is a smooth ${d}$-manifold, then

$\displaystyle \begin{array}{rcl} H^k_c(\tilde M)=H_{d-k}(\tilde M). \end{array}$

Bieri-Eckmann observed that is suffices that ${\Gamma}$ acts freely cocompactly on a contractible space ${X}$ whose compactly supported cohomology vanishes in all degrees but ${d}$, and ${H_c^d(X)}$ is torsion free. Then ${\Gamma}$ is a duality group.

If the action is merely proper and cocompact, ${\Gamma}$ 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)${, Bestvina-Feighn defined a bordification too. We proceed differently, like Grayson: we produce and invariant retract }$J_n\subset X_n$latex {. It is much easier and gives more information on the boundary. \section{Construction} }&fg=000000$X_n${ is the space of metric graphs (without separating edges) with homotopy markings. It is made of simplices with edge-lengths as coordinates. }$J_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).