The boundary of the outer space of a free product (1408.0543v3)

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.