The boundary of the outer space of a free product

TL;DR

This paper characterizes the boundary of the outer space associated with a free product of groups, describing its topology, dimension, and properties of certain trees related to the group structure.

Contribution

It identifies the closure of the outer space with a space of minimal, very small trees and computes its topological dimension and boundary properties.

Findings

Topological dimension of the outer space is 3N+2k-4.

Boundary of the outer space has dimension 3N+2k-5.

Any very small tree has at most 2N+2k-2 orbits of branch points.

Abstract

Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space $P\mathcal{O}(G,\{G_1,\dots,G_k\})$ for the axes topology with the space of projective minimal, \emph{very small} $(G,\{G_1,\dots,G_k\})$-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the $G_i$'s, and whose tripod stabilizers are trivial. Its topological dimension is equal to $3N+2k-4$, and the boundary has dimension $3N+2k-5$. We also prove that any very small $(G,\{G_1,\dots,G_k\})$-tree has at most $2N+2k-2$ orbits of branch points.