The boundary of the outer space of a free product

Abstract

Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast \cdots \ast G_k\ast F_N$, 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.

arXiv commentv3: Final version, to appear in the Israel Journal of Mathematics. Section 3, regarding the definition and properties of geometric trees, has been rewritten to improve the exposition, following a referee's suggestion