

a) Peerreviewed
 Growth Tight Actions w/
Goulnara
N. Arzhantseva and Jing
Tao, Pacific Journal of Mathematics, 278 (2015), no. 1,
1–49.
 A proper action of a group \(G\) on a metric space \(X\) is growth
tight if for every infinite index normal subgroup \(N\) of \(G\) the exponential growth rate of \(G\) with respect to \(X\),
ie, of an orbit of \(G\) in \(X\), is strictly greater than the
corresponding growth rate
of \(G/N\). We show that an action is growth tight if it has a
strongly contracting element and if there is an orbit that is
sufficiently convex. The latter condition is satisfied if there is a
quasiconvex orbit, but also in other interesting cases such as the
action of the mapping class group on Teichmüller space.
 Mapping Tori of Free Group Automorphisms, and the BieriNeumannStrebel Invariant of Graphs of Groups w/ Gilbert Levitt,
Journal of Group Theory, 19 (2016), no. 2, 191–216.
 We compute BieriNeumannStrebel (BNS) invariants for certain graphs
of groups and hierarchies of groups. In particular, we show that the
BNS invariant of the mapping torus of a polynomially growing free group
automorphism is the complement of finitely many rationally defined subspheres.
 Splitting Line Patterns in Free
Groups, Algebraic & Geometric Topology, 16
(2016), no. 2, 621–673.
 Techniques of Paper 2 are used to derive a canonical JSJ
decomposition of a free group \(\mathbb{F}\) relative to an
algebraic lamination \(\mathcal{L}\) in terms of the topology of the
quotient \(\partial\mathbb{F}/\partial\mathcal{L}\).
As an application, we characterize virtual geometricity (cf Paper
3) as having a relative JSJ decomposition with geometric pieces.
 Growth Tight Actions of Product
Groups w/ Jing
Tao, Groups, Geometry and
Dynamics, 10 (2016), no. 2, 753–770.
 If \(G=\prod_{i=1}^nG_i\) is a product of finitely generated
groups and for each \(i\) we have a proper, cocompact action of \(G_i\) on a
metric space \(X_i\) with a strongly contracting element then the product
action of \(G\) on \(X=\prod_{i=1}^nX_i\) is growth tight if \(X\)
is given the \(L^p\) metric for \(p>1\) and not growth tight if
\(X\) is given the \(L^1\) metric.
In particular, if the \(X_i\) are Cayley graphs of the \(G_i\) then
the \(L^1\) metric and the \(L^\infty\) metric both correspond to
word metrics on \(G\). This provides the first construction of
a group such that the action on one of its Cayley graphs is growth tight and
the action on another of its Cayley graphs is not.
 Quasiisometries Need Not Induce Homeomorphisms of Contracting
Boundaries with the Gromov Product Topology, Analysis and
Geometry in Metric Spaces, 4 (2016), no. 1, 278–281.

We consider a 'contracting boundary' of a proper geodesic
metric space consisting of equivalence classes of geodesic rays that
behave like geodesics in a hyperbolic space.
We topologize this set via the Gromov product, in analogy to
the topology of the boundary of a hyperbolic space.
We show that when the space is not hyperbolic,
quasiisometries do not necessarily give homeomorphisms of this
boundary.
Continuity can fail even when the spaces are required to be
CAT(0).
We show this by constructing an explicit example.
 Quasiisometries Between
Groups with TwoEnded Splittings w/ Alexandre Martin,
Mathematical Proceedings of the Cambridge Philosophical Society, in
press, 43pp.
 We construct invariants for boundary homeomorphism and
quasiisometry of hyperbolic groups that split over twoended
subgroups in terms of the respective homeomorphism/quasiisometry types of the vertex groups
relative to the edge groups. For boundary homeomorphism we get a
complete invariant. For quasiisometry we get a complete invariant
when the vertex groups are rigid relative to their incident edge
groups.
 Characterizations of Morse Quasigeodesics via
Superlinear Divergence and Sublinear Contraction w/ Goulnara
N. Arzhantseva, Dominik
Gruber, and David Hume, submitted, 29pp. preprint
 We initiate a systematic study of contracting projections. We
show that sublinear contraction is equivalent to the wellknown
Morse property and to the property of having completely superlinear
divergence. We prove sublinear analogues of several theorem about
strongly contracting geodesics.
 Contracting Geodesics in Graphical Small Cancellation Groups w/ Goulnara
N. Arzhantseva, Dominik
Gruber, and David Hume, submitted, 41pp. preprint
 We give necessary and sufficient conditions for a geodesic in a
graphical small cancellation group to be contracting. Some
consequences are:
 Many \(Gr'(1/6)\) graphical small
cancellation group contains a strongly contracting element.
 A geodesic \(\alpha\) in a classical \(C'(1/6)\) small cancellation
group is Morse if and only if there is a sublinearly growing
function \(f\) such that for every relator \(\Pi\) we have
\(\alpha\cap\Pi\leq f(\Pi)\).
We construct uncountably many quasiisometry classes of torsionfree,
finitely generated groups such that every element is strongly
contracting. The examples include Gromov monster groups.
We show that when the defining graph has finite components then
the translation lengths of group elements are rational and
bounded away from 0.
We also construct an example of a finitely generated group with a cyclic subgroup
that is strongly contracting in one Cayley graph but not in another.
b) Non peerreviewed
c) Standalone publications
d) Publications for the general public and other publications


