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).

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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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