Whitehead Graphs Research Articles

The visual boundary of hyperbolic free-by-cyclic groups

Let ϕ be an atoroidal outer automorphism of the free group Fn. We study the Gromov boundary of the hyperbolic group Gϕ = Fn ⋊ϕ ℤ. Using the Cannon-Thurston map, we explicitly describe a family of embeddings of the complete bipartite graph K3,3 into the boundary of the free-by-cyclic group. To do so, we define the directional Whitehead graph and use it to relate the topology of the boundary to the structure of the Rips Machine associated to a fully irreducible outer automorphism of the free group. In particular, we prove that an indecomposable Fn-tree is Levitt type if and only if one of its directional Whitehead graphs contains more than one edge. As an application, we obtain a new proof of Kapovich-Kleiner’s theorem [KK00] that ∂Gϕ is homeomorphic to the Menger curve if the automorphism is atoroidal and fully irreducible.

Random outer automorphisms of free groups: Attracting trees and their singularity structures

We prove that a random free group outer automorphism is an ageometric fully irreducible outer automorphism whose ideal Whitehead graph is a union of triangles. In particular, we show that its attracting (and repelling) tree is a nongeometric $\mathbb R$-tree all of whose branch points are trivalent

Planar Whitehead graphs with cyclic symmetry arising from the study of Dunwoody manifolds

A fundamental theorem in the study of Dunwoody manifolds is a classification of finite graphs on 2n vertices that satisfy seven conditions (concerning planarity, regularity, and a cyclic automorphism of order n). Its significance is that if the presentation complex of a cyclic presentation is a spine of a 3-manifold then its Whitehead graph satisfies the first five conditions (the remaining conditions do not necessarily hold). In this paper we observe that this classification relies implicitly on an unstated 8th condition and that this condition is not necessary for such a presentation complex to be the spine of a 3-manifold. We expand the scope of Dunwoody’s classification by classifying all graphs that satisfy the first five conditions.

Irreducible nonsurjective endomorphisms of Fn are hyperbolic

Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an irreducible immersion on a finite graph. We give a new proof of this and also show a partial converse holds when the immersion has connected Whitehead graphs with no cut vertices. The next result is a characterization of finitely generated subgroups of the free group that are invariant under an irreducible nonsurjective endomorphism. Consequently, irreducible nonsurjective endomorphisms are fully irreducible. The characterization and Reynolds' theorem imply that the mapping torus of an irreducible nonsurjective endomorphism is word-hyperbolic.

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [24] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an [Formula: see text] analog of the Masur–Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [16] give a strictly finer invariant in the analogous [Formula: see text] setting, we determine which of the 21 connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in [Formula: see text]. The rank 2 case is actually a direct consequence of the work of Masur and Smillie, as all elements of [Formula: see text] are induced by surface homeomorphisms and the index list and ideal Whitehead graph for a surface homeomorphism give precisely the same data.

Whitehead graphs and separability in rank two

Whitehead graphs and separability in rank two

Hyperbolic surface subgroups of one-ended doubles of free groups

Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank 2 or (2) every generator is used the same number of times in a minimal automorphic image of the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).

Ideal Whitehead graphs in $$\textit{Out}(F_r)$$ Out ( F r ) II: the complete graph in each rank

We show how to construct, for each \(r \ge 3\), an ageometric, fully irreducible \(\phi \in Out(F_r)\) whose ideal Whitehead graph is the complete graph on \(2r-1\) vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible \(\phi \in Out(F_3)\). The result is a first step to an \(Out(F_r)\) version of the Masur–Smillie theorem proving precisely which index lists arise from singular measured foliations for pseudo-Anosov mapping classes. In this paper we additionally give a method for finding periodic Nielsen paths and prove a criterion for identifying representatives of ageometric, fully irreducible \(\phi \in Out(F_r)\).

A Freiheitssatz for Whitehead Graphs of One-Relator Groups with Small Cancellation

In this work we study one-relator groups with a certain small cancellation condition. We focus on the following two general problems: the free subgroups of these groups, and what can be said on the automorphism group of these groups. Both problems are widely open. We introduce a graph-theoretical test which, if successful, implies that the subgroup under consideration is free. We also extend a result due to V. Shpilrain on automorphism groups of one-relator groups.

Whitehead Graphs and the Σ1-Invariants of Infinite Groups

Let G be a finitely generated group. The Bieri–Neumann–Strebel invariant Σ1(G) of G determines, among other things, the distribution of finitely generated subgroups N◃G with G/N abelian. This invariant can be quite difficult to compute. Given a finite presentation 〈S:R〉 for G, there is an algorithm, introduced by Brown and extended by Bieri and Strebel, which determines a space Σ(R) that is always contained in, and is sometimes equal to, Σ1(G). We refine this algorithm to one which involves the local structure of the universal cover of the standard 2-complex of a given presentation. Let Ψ(R) denote the space determined by this algorithm. We show that Σ(R) ⊆ Ψ ⊆ Σ1(G) for any finitely presented group G, and if G admits a staggered presentation, then Ψ = Σ1(G). By casting this algorithm in terms of connectivity properties of graphs, it is shown to be computationally feasible.