# Research

Here is my CV.

## Publications and Preprints

In this paper we develop the theory of train track maps on graphs of groups. We
prove the existence of CTs representing outer automorphisms of free products.
We generalize an index inequality due to Feighn-Handel to the graph of groups
setting, sharpening a result of Martino.

We introduce a quantitative characterization of subgroup alternatives
modeled on the Tits alternative
in terms of group laws
and investigate when this property is preserved under extensions.
We develop a framework that lets us expand the classes of groups
known to have locally uniform exponential growth to include
extensions of either word hyperbolic or right-angled Artin groups
by groups with locally uniform exponential growth.
From this, we deduce that the automorphism group
of a torsion-free one-ended hyperbolic group has locally uniform exponential growth.
Our methods also demonstrate that automorphism groups
of torsion-free one-ended toral relatively hyperbolic groups
and certain right-angled Artin groups satisfy our quantitative subgroup alternative.

In a seminal paper,
Stallings introduced folding of morphisms of graphs.
One consequence of folding is the representation of finitely-generated subgroups
of a finite-rank free group as immersions of finite graphs.
Stallings's methods allow one to construct this representation algorithmically,
giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups.
Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups,
which act geometrically on CAT(0) cube complexes.
In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

We show the existence
of several new infinite families of polynomially-growing automorphisms of free groups
whose mapping tori are CAT(0) free-by-cyclic groups.
Such mapping tori are thick, and thus not relatively hyperbolic.
These are the first families comprising infinitely many examples for each rank
of the nonabelian free group;
they contrast strongly with Gersten's example of a thick free-by-cyclic group
which cannot be a subgroup of a CAT(0) group.

Given a finite subgroup G of the mapping class group of a surface S,
the Nielsen realization problem asks whether G can be realized as a
finite group of homeomorphisms of S.
In 1983, Kerckhoff showed that for S a finite-type surface, any finite subgroup G
may be realized as a group of isometries of some hyperbolic metric on S.
We extend Kerckhoff's result to orientable, infinite-type surfaces.
As applications, we classify torsion elements in the mapping class group of a plane minus a Cantor set,
and also show that topological groups containing sequences of torsion elements limiting to the identity
do not embed continuously into the mapping class group of S.
Finally, we show that compact subgroups of the mapping class group of S are finite,
and locally compact subgroups are discrete.

**Train Tracks on Graphs of Groups and Outer Automorphisms of Hyperbolic Groups**

*PhD Thesis.* Available at: arXiv:2005.00164.

Abstract

Stallings remarked that an outer automorphism of a free group
may be thought of as a subdivision of a graph followed by a sequence of folds.
In this thesis, we prove that automorphisms of fundamental groups of graphs of groups
satisfying this condition may be represented by irrreducible train track maps
in the sense of Bestvina–Handel (we allow collapsing invariant subgraphs).
Of course, we construct relative train track maps as well.
Along the way, we give a new exposition of the Bass–Serre theory of groups acting on trees,
morphisms of graphs of groups, and foldings thereof.
We produce normal forms for automorphisms of free products and extend an argument
of Qing–Rafi to show that they are not quasi-geodesic.
As an application, we answer affirmatively a question of Paulin:
outer automorphisms of finitely generated word hyperbolic groups
satisfy a dynamical trichotomy generalizing the Nielsen–Thurston
"periodic, reducible or pseudo-Anosov."
At the end of the thesis we collect some open problems we find interesting.

## Geometric Group Theory without Boundaries

Along with Macarena Arenas, Sami Douba and Abdul Zalloum,
I organized a virtual “summer school” that ran July 2020.
More info is available on the summer school website.

I gave three lectures in this summer school on Bass and Serre’s theory
of graphs of groups and groups acting on trees.

Here are notes (with the annotations I added live)
for the lectures, as well as links to recordings of the lectures.