A compactification of outer space which is an absolute retract

TL;DR

The paper introduces the Pacman compactification of outer space, which is an absolute retract with a boundary that is a Z-set, improving the topological properties of the classical compactification.

Contribution

It defines a new compactification of outer space that is an absolute retract and has a boundary as a Z-set, addressing limitations of the classical compactification.

Findings

Pacman compactification is an absolute retract.

Boundary of the compactification is a Z-set.

Classical compactification fails to be locally 4-connected for N≥4.

Abstract

We define a new compactification of outer space $CV_N$ (the \emph{Pacman compactification}) which is an absolute retract, for which the boundary is a $Z$-set. The classical compactification $\overline{CV_N}$ 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 $\overline{CV_N}$, obtained by assigning an orientation to every arc with nontrivial stabilizer in the trees.