Boundary of the Relative Outer Space

TL;DR

This paper investigates the boundary structure of the relative outer space associated with free factors of a free group, establishing that its boundary has dimension exactly one less than the space itself.

Contribution

It proves that the boundary of the relative outer space has dimension one less than the space, extending understanding of the geometric structure of these automorphism groups.

Findings

Boundary dimension equals space dimension minus one.

Provides new insights into the topology of relative outer spaces.

Enhances understanding of automorphism group actions on free groups.

Abstract

Let $\mathcal{A} = {A_1, ..., A_k}$ be a system of free factors of $F_n$. The group of relative automorphisms $\mathrm{Aut}(F_n; \mathcal{A})$ is the group given by the automorphisms of $F_n$ that restricted to each $A_i$ are conjugations by elements in $F_n$. The group of relative outer automorphisms is defined as $\mathrm{Out}(F_n; \mathcal{A}) = \mathrm{Aut}(F_n; \mathcal{A}) / \mathrm{Inn}(F_n)$, where $\mathrm{Inn (F_n)$ is the normal subgroup of $\mathrm{Aut}(F_n)$ given by all the inner automorphisms. This group acts on the relative outer space $\mathrm{CV}_n(\mathcal{A})$. We prove that the dimension of the boundary of the relative outer space is $\mathrm{dim}(\mathrm{CV}_n(\mathcal{A}))-1$.