Limit trees for free group automorphisms: universality

To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique expanding dilation of the tree that is equivariant with respect to the automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to topmost attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism's topmost exponential dynamics. As a corollary, our previously constructed limit pretree that detect the exponential dynamics is canonical; the pretree admits pseudometrics that can be viewed as a universal hierarchy of limit trees. For atoroidal automorphisms, this universal hierarchical decomposition is analogous to the Nielsen--Thurston normal form for a surface homeomorphism or the Jordan canonical form for a linear map. We use it to sketch a proof that atoroidal outer automorphisms have virtually abelian centralizers.Comment: 58 pages, 3 figure

Hyperbolic Endomorphisms of Free Groups

We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends Brinkmann\u27s theorem that free-by-cyclic groups are word-hyperbolic if and only if they have no Z2 subgroups. To get started on our main theorem, we first prove a structure theorem for injective but nonsurjective endomorphisms of free groups. With the decomposition of the free group given by this structure theorem, we (more or less) construct representatives for nonsurjective endomorphisms that are expanding immersions relative to a homotopy equivalence. This structure theorem initializes the development of (relative) train track theory for nonsurjective endomorphisms

The minimal genus problem for right angled Artin groups

We investigate the minimal genus problem for the second homology of a right angled Artin group (RAAG). Firstly, we present a lower bound for the minimal genus of a second homology class, equal to half the rank of the corresponding cap product matrix. We show that for complete graphs, trees, and complete bipartite graphs, this bound is an equality, and furthermore in these cases the minimal genus can always be realised by a disjoint union of tori. Additionally, we give a full characterisation of classes that are representable by a single torus. However, it is not true in general that the minimal genus of a second homology class of a RAAG is necessarily realised by a disjoint union of tori: we construct a genus two representative for a class in the pentagon RAAG.Comment: 19 pages, 4 figures; comments welcom

Hyperbolic hyperbolic-by-cyclic groups are cubulable

We show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Along the way, we (i) give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable, and (ii) extend to the case with torsion Brinkmann's thesis that a torsion-free hyperbolic-by-cyclic group is hyperbolic if and only if it does not contain $\mathbb{Z}^2$-subgroups.Comment: 11 page

Limit pretrees for free group automorphisms: existence

To any free group automorphism, we associate a real pretree with several nice properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism of the real pretree that represents the free group automorphism. Finally and crucially, the loxodromic elements are exactly those whose (conjugacy class) length grows exponentially under iteration of the automorphism; thus, the action on the real pretree is able to detect the growth type of an element. This construction extends the theory of metric trees that has been used to study free group automorphisms. The new idea is that one can equivariantly blow up an isometric action on a real tree with respect to other real trees and get a rigid action on a treelike structure known as a real pretree. Topology plays no role in this construction as all the work is done in the language of pretrees.Comment: 41 pages, 1 figure. v2: 40 pages, referenced the seque

Hyperbolic hyperbolic-by-cyclic groups are cubulable

11 pagesWe show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Along the way, we (i) give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable, and (ii) extend to the case with torsion Brinkmann's thesis that a torsion-free hyperbolic-by-cyclic group is hyperbolic if and only if it does not contain $\mathbb{Z}^2$-subgroups