Limit pretrees for free group automorphisms: existence, preprint.(abstractabstract, pdf)
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. Secondly, 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. Finally and crucially, there is an expanding pretree-automorphism of the real pretree that is equivariant with respect to the free group automorphism.
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.
abstractabstract, pdf, arxiv)
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.
Constructing stable images, submitted.(abstractabstract, pdf, mpim)
There is an algorithm for constructing a canonical representative for an injective free group endomorphism. The main corollary to our algorithm is an affirmative answer to Ventura's question: yes, the stable image for a free group endomorphism can be computed. This corollary also generalizes to all finite rank free groups a result due to Ciobanu–Logan in rank 2. By work of Bogopolski–Maslakova, it implies that the fixed point subgroup of a free group endomorphism can be computed. The final corollary is that the hyperbolicity of an ascending HNN extension of a free group can be algorithmically determined by looking solely at the dynamics of the defining monodromy.
The dynamics and geometry of free group endomorphisms, Adv. Math. 384 (2021) 107714.(abstractabstract, pdf, arxiv, slides)
We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends the theorem of Brinkmann that free-by-cyclic groups are word-hyperbolic if and only if they have no free abelian subgroups of rank 2. The paper is split into two independent parts:
- We study the dynamics of injective nonsurjective endomorphisms of free groups. We prove a canonical structure theorem that initializes the development of improved relative train tracks for endomorphisms; this structure theorem is of independent interest since it makes many open questions about injective endomorphisms tractable.
- As an application of the structure theorem, we are able to (relatively) combine Brinkmann's theorem with our previous work and obtain the main result stated above. In the final section, we further extend the result to HNN extensions of free groups over free factors.
Remark. The published version has a different section numbering from the preprint. The sections 1-7 changed to subsections 3.1-3.4 and 5.1-5.3, respectively.
Irreducibility of a Free Group Endomorphism is a Mapping Torus Invariant, Comment. Math. Helv. 96 (2021) 47–63.(abstractabstract, pdf, arxiv)
We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index subgroups that are ascending HNN extensions.
Irreducible Nonsurjective Endomorphisms of \(F_n\) are Hyperbolic, Bull. Lond. Math. Soc. 52 (2020) 960–976.(abstractabstract, pdf, arxiv, blog)
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.
Hyperbolic Immersions of Free Groups, Groups Geom. Dym. 14 (2020) 1253–1275.(abstractabstract, pdf, arxiv)
We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag-Solitar subgroups. Due to a result by Reynolds, this theorem applies to all injective endomorphisms of \(F_2\) and nonsurjective fully irreducible endomorphisms of \(F_n\). We also give a framework for extending the theorem to all injective endomorphisms of \(F_n\).