When is the Outer Space of a free product CAT(0)?

TL;DR

This paper investigates when the Outer Space for free products of groups admits a CAT(0) metric, providing a complete characterization and revealing mostly negative results, with a notable exception in a specific two-dimensional case.

Contribution

It characterizes conditions under which the Outer Space for free products supports a CAT(0) metric, extending previous work and identifying a unique two-dimensional case that does support such a metric.

Findings

Most high-dimensional cases are not CAT(0)

A specific two-dimensional free product Outer Space is CAT(0)

Extends previous thesis work on Outer Space geometry

Abstract

Generalizing Culler and Vogtmann's Outer Space for the free group, Guirardel and Levitt construct an Outer Space for a free product of groups. We completely characterize when this space (or really its simplicial spine) supports an equivariant piecewise-Euclidean or piecewise-hyperbolic CAT(0) metric. Our results are mostly negative, extending thesis work of Bridson and related to thesis work of Cunningham. In particular, provided the dimension of the spine is at least three, it is never CAT(0). Surprisingly, we exhibit one family of free products for which the Outer Space is two-dimensional and does support an equivariant CAT(0) metric.