End invariants of the group of outer automorphisms of a free group
It is known that every finitely generated group has 0, 1, 2 or infinitely many ends. If the group is finitely presented, then the number of ends is equal to the number of ends of any simply connected complex on which the group acts freely with finite quotient. If the ends are simply connected the group is said to be simply connected at infinity (a property which is independent of the complex). The author considers the group \(\text{Out}(F_n)\) of outer automorphisms of a free group of rank \(n\). The author proves that this group has infinitely many ends for \(n=2\) and one end for \(n\geq 3\), but the main result of the paper is that \(\text{Out}(F_n)\) is simply connected at infinity for \(n\geq 5\). The proof is carried out by considering the ``outer space'' \(X_n\) of homothety classes of free minimal actions of \(F_n\) on simplicial \(\mathbb{R}\)-trees, defined in a paper by \textit{M. Culler} and \textit{K. Vogtmann} [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)].
scientific article; zbMATH DE number 794438