Free Splitting Complex Research Articles

Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification – the free splitting complex – with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of A u t ( F n ) Aut(F_n) is a well-ordered subset of R \mathbb {R} , this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call partial train tracks (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement – minpoints – coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.

Homotopy type of the complex of free factors of a free group

We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n-2)-connected. In addition, we show that for every non-trivial free factor system of a free group, the corresponding relative free splitting complex is contractible.

Panel collapse and its applications

We describe a procedure called panel collapse for replacing a CAT(0) cube complex \Psi by a "lower complexity" CAT(0) cube complex \Psi_\bullet whenever \Psi contains a codimension-2 hyperplane that is extremal in one of the codimension-1 hyperplanes containing it. Although \Psi_\bullet is not in general a subcomplex of \Psi , it is a subspace consisting of a subcomplex together with some cubes that sit inside \Psi "diagonally." The hyperplanes of \Psi_\bullet extend to hyperplanes of \Psi . Applying this procedure, we prove: if a group G acts cocompactly on a CAT(0) cube complex \Psi , then there is a CAT(0) cube complex \Omega so that G acts cocompactly on \Omega and for each hyperplane H of \Omega , the stabiliser in G of H acts on H essentially. Using panel collapse, we obtain a new proof of Stallings's theorem on groups with more than one end. As another illustrative example, we show that panel collapse applies to the exotic cubulations of free groups constructed by Wise in [44]. Next, we show that the CAT(0) cube complexes constructed by Cashen and Macura in [7] can be collapsed to trees while preserving all of the necessary group actions. (It also illustrates that our result applies to actions of some non-discrete groups.) We also discuss possible applications to quasi-isometric rigidity for certain classes of graphs of free groups with cyclic edge groups. Panel collapse is also used in forthcoming work of the first-named author and Wilton to study fixed-point sets of finite subgroups of Out (F_n) on the free splitting complex. Finally, we apply panel collapse to a conjecture of Kropholler, obtaining a short proof under a natural extra hypothesis.

Loxodromics for the cyclic splitting complex and their centralizers

We show that an outer automorphism acts loxodromically on the cyclic splitting complex if and only if it has a filling lamination and no generic leaf of the lamination is carried by a vertex group of a cyclic splitting. This is the analog for the cyclic splitting complex of Handel-Mosher's theorem on loxodromics for the free splitting complex. We also show that such outer automorphisms have virtually cyclic centralizers.

The metric completion of Outer Space

We develop the theory of a metric completion of an asymmetric metric space. We characterize the points on the boundary of Outer Space that are in the metric completion of Outer Space with the Lipschitz metric. We prove that the simplicial completion, the subset of the completion consisting of simplicial tree actions, is homeomorphic to the free splitting complex. We use this to give a new proof of a theorem by Francaviglia and Martino that the isometry group of Outer Space is isomorphic to $${\text {Out}}(F_{n})$$ for $$n \ge 3$$ and to $$\text {PSL}(2,{{\mathbb {Z}}})$$ for $$n=2$$.

The free splitting complex of a free group, II: Loxodromic outer automorphisms

We study the loxodromic elements for the action of O u t ( F n ) \mathsf {Out}(F_n) on the free splitting complex of the rank n n free group F n F_n . Each outer automorphism is either loxodromic or has bounded orbits without any periodic point, or has a periodic point; all three possibilities can occur. Two loxodromic elements are either coaxial or independent, meaning that their attracting-repelling fixed point pairs on the Gromov boundary of the free splitting complex are either equal or disjoint as sets. Each alternative is characterized in terms of attracting laminations; in particular, an outer automorphism is loxodromic if and only if it has a filling attracting lamination. As an application, each attracting lamination determines its corresponding repelling lamination independent of the outer automorphism. As part of this study, we describe the structure of the subgroup of O u t ( F n ) \mathsf {Out}(F_n) that stabilizes the fixed point pair of a given loxodromic outer automorphism, and we give examples which show that this subgroup need not be virtually cyclic. As an application, the action of O u t ( F n ) \mathsf {Out}(F_n) on the free splitting complex is not acylindrical, and its loxodromic elements do not all satisfy the weak proper discontinuity (WPD) property of Bestvina and Fujiwara.

Nielsen Realization by Gluing: Limit Groups and Free Products

We generalize the Karrass–Pietrowski–Solitar and the Nielsen realization theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel and Mosher and on the outer space of a free product of Guirardel and Levitt, and also a relative version of the Nielsen realization theorem, which, in the case of free groups, answers a question of Karen Vogtmann. We also prove Nielsen realization for limit groups and, as a byproduct, obtain a new proof that limit groups are CAT(0). The proofs rely on a new version of Stallings’ theorem on groups with at least two ends, in which some control over the behavior of virtual free factors is gained.

The hyperbolicity of the sphere complex via surgery paths

Abstract In [10], Handel and Mosher have proved that the free splitting complex ℱ ⁢ 𝒮 n {\mathcal{F\kern-2.27622ptS}_{n}} for the free group F n {F_{n}} is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes ℱ ⁢ 𝒮 n {\mathcal{F\kern-2.27622ptS}_{n}} as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher’s sphere complex (another model for the free splitting complex), instead of Handel and Mosher’s fold paths. As a byproduct, we get that surgery paths are unparameterized quasi-geodesics in the sphere complex. We explain how to deduce from our proof the hyperbolicity of the free factor complex and the arc complex of a surface with boundary.

Uniform hyperbolicity of the curve graph via surgery sequences

We prove that the curve graph $\calC^{(1)}(S)$ is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface $S$. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.

On hyperbolicity of free splitting and free factor complexes

We show how to derive hyperbolicity of the free factor complex of F_N from the Handel–Mosher proof of hyperbolicity of the free splitting complex of F_N , thus obtaining an alternative proof of a theorem of Bestvina–Feighn. We also show that under the natural map \tau from the free splitting complex to free factor complex, a geodesic [x,y] maps to a path that is uniformly Hausdorff-close to a geodesic [\tau(x),\tau(y)] .

On the geometry of the edge splitting complex

The group Out of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which Out acts, in analogy with the curve complex for the mapping class group. Here, we focus on one of these proposed analogues: the edge splitting complex \mathcal{E}\mathcal{S}_n , equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton \mathcal{E}\mathcal{S}_n^1 algebraically, and use our characterization to find lower bounds on distances between points in this graph. Our distance calculations allow us to find quasiflats of arbitrary dimension in \mathcal{E}\mathcal{S}_n . This shows that \mathcal{E}\mathcal{S}_n : is not hyperbolic, has infinite asymptotic dimension, and is such that every asymptotic cone is infinite dimensional. These quasiflats contain an unbounded orbit of a reducible element of Out. As a consequence, there is no coarsely Out-equivariant quasiisometry between \mathcal{E}\mathcal{S}_n and other proposed curve complex analogues, including the regular free splitting complex \mathcal{F}\!\mathcal{S}_n , the (nontrivial intersection) free factorization complex \mathcal{F\!F}_n , and the free factor complex \mathcal{F}_n .

The free splitting complex of a free group, I Hyperbolicity

We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.