**One-endedness of outer automorphism groups of free products of finite and cyclic groups**

Available at arXiv:2305.04986.
Abstract

In a previous paper, we showed that the group of outer automorphisms of the free product
of two nontrivial finite groups with an infinite cyclic group has infinitely many ends,
despite being of virtual cohomological dimension two.
The main result of this paper is that aside from this exception,
having virtual cohomological dimension at least two implies the outer automorphism group
of a free product of finite and cyclic groups is one ended.
As a corollary, the outer automorphism groups of the free product of four finite groups
or the free product of a single finite group with a free group of rank two are
virtual duality groups of diimension two, in contrast with the above example.
We also prove that groups in this family are semistable at infinity (or at each end).
Our proof is inspired by methods of Vogtmann, applied to a complex first studied
in another guise by Krstić and Vogtmann.

**When is the Outer Space of a free product CAT(0)?**

Available at arXiv:2209.04711.
Abstract

Generalizing Culler and Vogtmann’s Outer Space for the free group,
Guirardel and Levitt construct an Outer Space for a free product of groups.
We completely characterize when this space (or really its simplicial spine)
supports an equivariant piecewise-Euclidean or piecewise-hyperbolic CAT(0) metric.
Our results are mostly negative, extending thesis work of Bridson and related to thesis work of Cunningham.
In particular, provided the dimension of the spine is at least three, it is never CAT(0).
Surprisingly, we exhibit one family of free products for which the Outer Space is two-dimensional
and _does_ support an equivariant CAT(0) metric.

**On Whitehead’s cut vertex lemma**

Available at arXiv:2205.06071.
Abstract

*J. Group Theory.*

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.

**CTs for free products**

Available at arXiv:2203.08868.
Abstract

*Submitted.*

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.

**Train track maps on graphs of groups**

Available at arXiv:2102.02848.
Abstract

*Groups, Geom. Dyn.* 16 (2022), no. 4, pp. 1389-1422

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.

**Extensions of hyperbolic groups have locally uniform exponential growth**

(with Robert Kropholler and Thomas Ng)

Available at arXiv:2012.14880.
Abstract

*Submitted.*

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.

**Folding-like techniques for CAT(0) cube complexes**

(with Michael Ben-Zvi and Robert Kropholler)

Available at arXiv:2011.05374.
Abstract

*Math. Proc. Camb. Philos. Soc.* 173(1) (2022), 227-238.

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.

**Some new CAT(0) free-by-cyclic groups**

Available at arXiv:1909.03097.
Abstract

*Mich. Math. J.* 73 (2023), 621–630

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.

**Nielsen realization for infinite-type surfaces**

(with Santana Afton, Danny Calegari and Lvzhou Chen)

Available at: arXiv:2002.09760.
Abstract

*Proc. Amer. Math. Soc.* 149 (2021), 1791–1799

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.