# Research

Here is my CV.

## Publications and Preprints

Consider the free product of two nontrivial finite groups with an infinite cyclic group.
We prove that the 2-dimensional spine of Outer Space for this free product
supports an equivariant CAT(0) metric with infinitely many ends.
The outer automorphism group of this free product is thus relatively hyperbolic.
In the special case that both finite groups are cyclic of order two,
we show that the outer automorphism group is virtually a certain Coxeter group,
and that the spine of Outer Space may be identified with its Davis–Moussong complex.
These outer automorphism groups thus exhibit behavior extremely different from
outer automorphism groups of free groups,
and conjecturally, from other outer automorphism groups of free products
of finite and cyclic groups.

One version of Whitehead's famous cut vertex lemma says that if an element of a free group
is part of a free basis, then a certain graph associated to its conjugacy class
that we call the star graph
is either disconnected or has a cut vertex.
We state and prove a version of this lemma for conjugacy classes of elements
and convex-cocompact subgroups of groups acting cocompactly on trees
with finitely generated edge stabilizers.

**Lipschitz metric isometries between Outer Spaces of virtually free groups**

Available at arXiv:2203.09008.
Abstract

*Submitted.*

Dowdall and Taylor observed that given a finite-index subgroup of a free group,
taking covers induces an embedding
from the Outer Space of the free group to the Outer Space of the subgroup,
that this embedding is an isometry with respect to the (asymmetric) Lipschitz metric,
and that the embedding sends folding paths to folding paths.
The purpose of this note is to extend this result to virtually free groups.
We further extend a result Francaviglia and Martino,
proving the existence of “candidates” for the Lipschitz distance
between points in the Outer Space of the virtually free group.
Additionally we identify a deformation retraction of the spine
of the Outer Space for the virtually free group with the space considered by Krstic and Vogtmann.

The fundamental group of a finite graph of groups with trivial edge groups is a free product.
We are interested in those outer automorphisms of such a free product
that permute the conjugacy classes of the vertex groups.
We show that in particular cases of interest,
such as where the vertex groups are themselves finite free products of finite and cyclic groups,
given such an outer automorphism,
after passing to a positive power, the outer automorphism
is represented by a particularly nice kind of relative train track map called a CT.
CTs were first introduced by Feighn and Handel
for outer automorphisms of free groups.
We develop the theory of attracting laminations for
and principal automorphisms of free products.
We prove that outer automorphisms of free products satisfy an index inequality
reminiscent of a result of Gaboriau, Jaeger, Levitt and Lustig
and sharpening a result of Martino.
Finally, we prove a result reminiscent of a result of Culler
on the fixed subgroup of an automorphism of a free product
whose outer class has finite order.

In this paper we develop the theory of train track maps on graphs of groups.
Expanding a definition of Bass, we define a notion of a map of a graph of groups,
and of a homotopy equivalence.
We prove that under one of two technical hypotheses,
any homotopy equivalence of a graph of groups may be represented by a relative train track map.
The first applies in particular to graphs of groups with finite edge groups,
while the second applies in particular to certain generalized Baumslag–Solitar groups.

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**

Available at: arXiv:2005.00164.
Abstract

*PhD Thesis.*

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.