1. On polynomial free-by-cyclic groups, preprint.

(abstractabstract, pdf)

A free-by-cyclic group can often be viewed as a mapping torus of a free group automorphism (monodromy) in multiple ways. What dynamical properties must these monodromies share, and to what extent are they invariant under quasi-isometries? We give a new proof using cyclic splittings that the polynomial growth of a monodromy is a geometric invariant of the free-by-cyclic group; we also characterise the degree of polynomial growth using slender splittings. For exponential growth, we conjecture that the nesting of attracting laminations is a geometric invariant.

(abstractabstract, pdf)

This survey is based on a minicourse I taught at UC-Riverside in March 2023. The goal is to show the relation between the dynamics of a free group endomorphism and the geometry of its mapping torus. As a new result, we characterize the Dehn functions of ascending HNN extensions of free groups: they are linear, quadratic, or exponential.

3. Hyperbolic hyperbolic-by-cyclic groups are cubulable, with François Dahmani and Suraj Krishna M S, to appear in Geom. Topol.

(abstractabstract, pdf, arxiv)

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.

(abstractabstract, pdf, arxiv)

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 represents the free group automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism's dominating exponential dynamics.

As a corollary, our previously constructed limit pretree that detects the exponential dynamics is canonical. We also characterize all very small trees that admit an expanding homothety representing a given automorphism. In the appendix, we prove a variation of Feighn--Handel's recognition theorem for atoroidal outer automorphisms.

5. Limit pretrees for free group automorphisms: existence, Forum Math. Sigma 12 (2024) e57.

(abstractabstract, pdf, arxiv, slides)

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 (intervals).

6. The minimal genus problem for right angled Artin groups, with Rachael Boyd and Thorben Kastenholz, Geom. Dedicata 217 (2023) 93.

(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, the minimal genus of a second homology class of a RAAG is not always realised by a disjoint union of tori as an example we construct in the pentagon shows.

7. Constructing stable images, preprint.

(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.

8. 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:

1. 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.
2. 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.

9. 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.

10. 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.

11. 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 $$F_n$$.

Dissertation: Hyperbolic Endomorphisms of Free Groups. (scholarworks, defense)

Note: "The dynamics and geometry of free group endomorphisms" is a restructured-for-publication version of my dissertation with a few extra results.