Sphere paths in outer space

Abstract

We give estimates on the length of paths defined in the sphere model of outer space using a surgery process, and show that they make definite progress in some sense when they remain in some thick part of outer space. To do so, we relate the Lipschitz metric on outer space to a notion of intersection numbers. 20F65 IntroductionIn order to study the outer automorphism group of a finitely generated free group, Culler and Vogtmann introduced a space, called outer space, on which the group Out.F n / acts in a nice way (see Culler and Vogtmann [4;13].) This space is built as an analog of Teichmüller spaces, used to study the mapping class group of a surface. While Teichmüller spaces are equipped with several interesting metrics, whose properties have been investigated a lot, there had been no systematic investigation of metric properties of outer space before Francaviglia and Martino studied an analog of Thurston's asymmetric metric [5]. In particular, Francaviglia and Martino proved that outer space is geodesic for this metric, the geodesics being obtained by using a folding process.Building on ideas of Whitehead [14], Hatcher defined a new model for outer space, using sphere systems in a 3-dimensional manifold with fundamental group F n [10]. In order to prove the contractibility of the full sphere complex, he also defined a combing path in this model of outer space, which appears to look like an "unfolding path". A modification of this path was also used by Hatcher and Vogtmann to prove exponential isoperimetric inequalities for Out.F n / [11].Our goal is to investigate the metric properties of this path. As combing paths are piecewise linear, we can talk about their vertices, and define the length l. / of a combing path to be the sum of the distances from one vertex to the next. We prove the following result.

An algorithm to detect full irreducibility by bounding the volume of periodic free factors

“…Lemma 8 (cf. Proposition 2.8 in [15]). Suppose S, T ∈ cv are unit roses and that f : S → T is a length minimizing morphism.…”

“…Often we may assume S and T are unit roses, in which case i(S, T ) simply counts the squares in C(S × T )/F . For our purpose, we do not need the full definition of C(S × T ), for which we refer the reader to [11,15]. Rather, we make use of two approaches to computing i(S, T ).…”

“…Horbez details the correspondence between geometric intersection of the sphere systems and the volume of the Guirardel core [15]. In particular he shows that, if T 0 , T 1 are trees in CV , then for the corresponding sphere systems Σ 0 , Σ 1 ∈ S − S ∞ , we have i(T 0 , T 1 ) = i(Σ 0 , Σ 1 ), where the latter counts the minimal number of circles common to each sphere system, weighted appropriately.…”

Patterson–Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for outer space

“…Using the result of Horbez [32] stated in ( ‡) above, Theorem 1.1 directly implies (using the notation introduced after ( ‡)):…”

“…The set cv 1 N,ε is called the ε-thick part of cv 1 N . Horbez [32] showed that, for any fixed ε > 0, if T, S ∈ cv 1 N,ε , one has…”