A compactification of outer space which is an absolute retract

Abstract

We define a new compactification of outer space $C{V}_N$ (the \emph{Pacman compactification}) which is an absolute retract, for which the boundary is a $Z$-set. The classical compactification $\overline{C{V}_N}$ made of very small $F_N$-actions on $\mathbb{R}$-trees, however, fails to be locally $4$-connected as soon as $N\ge 4$. The Pacman compactification is a blow-up of $\overline{C{V}_N}$, obtained by assigning an orientation to every arc with nontrivial stabilizer in the trees.

arXiv commentFinal version. To appear in Annales de l'Institut Fourier