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 Counting in groups: Fine asymptotic geometry,
Notices of the AMS 63, No. 8 (2016), 871874.
(Published version; see also
Cover and
Explanation)
Short expository treatment of rational growth in groups and its connection to geometric counting problems like Ehrhart quasipolynomials
and the Gauss circle problem.

 Rational growth in the Heisenberg group (with Mike Shapiro),
submitted.
(math.GR/1411.4201,
most recent)
We show that the standard Heisenberg group \(H(\mathbb{Z})\) has rational growth with respect to any
finite generating set. Since the growth is bounded by a polynomial, this is the same as saying that
the growth function is eventually quasipolynomial. Previously, rational growth in any generators
was only known for hyperbolic groups and virtually abelian groups. This settles a longstanding open
question.

 Random nilpotent groups I
(with Matt Cordes,
Yen Duong,
Turbo Ho, and
Andrew Sánchez),
IMRN, to appear.
(math.GR/1506.01426)
We study random nilpotent groups in the wellestablished style of random groups, by choosing relators uniformly among freely reduced words of (nearly) equal length and letting the length tend to infinity. Whereas random groups are quotients of a free group by such a random set of relators, random nilpotent groups are formed as corresponding quotients of a free nilpotent group. This model reveals new phenomena because nilpotent groups are not "visible" in the standard model of random groups (due to the sharp phase transition from infinite hyperbolic to trivial groups).

 A sharper threshold for random groups at density onehalf
(w/
Kasia Jankiewicz,
Shelby Kilmer,
Samuel Lelièvre,
John Mackay, and
Andrew Sánchez),
Groups, Geometry, and Dynamics Groups 10, No. 3 (2016), 9851005.
(math.GR/1412.8741)
In the density model of random groups, we consider presentations with any fixed number \(m\) of generators and many random relators of length \(\ell\),
sending \(\ell\) to infinity.
If \(d\) is a "density" parameter measuring the rate of exponential growth of the number of relators compared to the length of relators,
then many grouptheoretic properties become generically true or generically false at different values of \(d\).
The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for \(d < 1/2\),
random groups are a.a.s. infinite hyperbolic, while for \(d > 1/2\),
random groups are a.a.s. order one or two.
We study random groups at the density threshold \(d = 1/2\).
Kozma had found that trivial groups are generic for a range of growth rates at \(d = 1/2\);
we show that infinite hyperbolic groups are generic in a different range. (We include an exposition of Kozma's previously unpublished argument,
with slightly improved results, for completeness.)

 Equations in nilpotent groups (with Hao
Liang and Mike
Shapiro),
Proceedings of the AMS 143 (2015), 47234731.
(math.GR/1401.2471)
We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time.
The method works for all twostep nilpotent groups with rankone commutator, which includes the higher Heisenberg groups.
We also prove that the decision problem for systems of equations is unsolvable in all nonabelian free nilpotent groups.

 Statistical hyperbolicity in Teichmüller space
(with S
Dowdall and
H Masur),
GAFA,
Volume 24, Issue 3 (2014), 748795.
(math.GR/1108.5416;
Most recent)
In this paper we explore the idea that Teichmüller space with the Teichmüller metric is hyperbolic "on average."
We consider several different measures on Teichmüller space 
coming from the metric, the symplectic structure, the Finsler structure, and from quadratic differentials 
and show that with respect to each one,
the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible.
In fact, we prove a stronger result: a random triangle formed by sampling three points in the ball
will have the midpoints of all three sides fall within bounded distance of the center. This is a kind of
generic thin triangles result.

 Spheres in the curve complex
(with Spencer
Dowdall and
Howard Masur),
In the Tradition
of Ahlfors and Bers, Contemp. Math. 590 (2013), 18.
(math.GT/1109.6338;
Most recent)
In this short paper we study the typical or generic metric properties
in the curve complex of a surface.
The curve graph is locally infinite and does not support any invariant probability measures, so to
make sense of the idea of averaging, we
develop definitions of null and generic subsets in a way that is compatible with the topological structure of the curve complex.
With respect to this notion of genericity, we show that the concatenation of two geodesics with a common endpoint
is almost surely itself geodesic. In this sense the curve graph might be said to be "even more hyperbolic than a tree."

 Fine asymptotic geometry in the Heisenberg group (with
Christopher Mooney),
Indiana Univ. Math Journal
63 No. 3 (2014), 885916.
(math.GR/1106.5276,
Most recent)
For every finite generating set on the integer Heisenberg group H(Z),
Pansu showed that the word metric has the largescale structure of a CarnotCarathéodory Finsler metric on the real
Heisenberg group H(R).
We study the properties of those limit metrics and obtain results about the geometry of word metrics that
reflect the dependence on generators.
For example we compute the probability that a group element has all of its geodesic spellings sublinearly close together,
relative to word length.
In free abelian groups of rank at least two, that probability is 0;
in unbounded hyperbolic groups, the probability is 1.
In H(Z) it is a rational number strictly between 0 and 1 that depends on the generating set;
with respect to the standard generators,
the probability is precisely 19/31.

 The geometry of spheres in free abelian groups
(with
Samuel Lelièvre and
Christopher Mooney),
Geom. Dedicata, Vol 161, Issue 1 (2012), 169187.
(math.GR/1004.0053;
More recent)
We study word metrics on Z^{d}
by developing tools that are ne enough
to measure dependence on the generating set. We obtain counting and distribution
results for the words of length n. With this, we show that counting measure
on spheres always converges to cone measure on a polyhedron (strongly, in an
appropriate sense). Using the limit measure, we can reduce probabilistic questions
about word metrics to problems in convex geometry of Euclidean space. We give
several applications to the statistics of "sizelike" functions.

 Statistical hyperbolicity in groups (with
Samuel Lelièvre and
Christopher Mooney),
Algebraic and Geometric Topology 12 (2012) 118.
(math.GR/1104.4460;
Most recent)
We introduce a geometric statistic called the sprawl of a group with
respect to a generating set, based on the average distance in the word metric
between pairs of words of equal length. The sprawl quantifies a certain obstruction
to hyperbolicity. Group presentations with maximum sprawl (i.e., without this
obstruction) are called statistically hyperbolic. We first relate sprawl to curvature
and show that nonelementary hyperbolic groups are statistically hyperbolic, then
give some results for products and for certain solvable groups. In free abelian
groups, the word metrics are asymptotic to norms induced by
convex polytopes,
causing several kinds of group invariants to reduce to problems in convex geometry.

 The sprawl conjecture for convex bodies
(with
Samuel Lelièvre and
Christopher Mooney),
Experimental Mathematics, Volume 22, Issue 2 (2013),
113122. (PDF)
In this short companion paper to the previous paper, we give some rigorous
and some empirical evidence for the "Sprawl Conjecture," that the
average distance between points on a sphere satisfies affine isoperimetric inequalities.

 Pushing fillings in rightangled Artin groups
(with
Aaron Abrams,
Noel Brady,
Pallavi Dani, and
Robert
Young),
Journal of the LMS, Vol 87, Issue 3 (2013), 663688.
(math.GR/1004.4253;
Most recent)
We define a family of quasiisometry invariants of groups called higher divergence functions, which measure
isoperimetric properties "at infinity." We give sharp upper and lower bounds on the divergence functions for
rightangled Artin groups, using different pushing maps on the associated cube complexes. In the process, we define
a class of RAAGs we call orthoplex groups, which have the property that their BestvinaBrady subgroups have
hardtofill spheres. Other results include sharp bounds on the higher Dehn functions of BestvinaBrady groups, and a
complete characterization of the divergence of geodesics in RAAGs.

 Filling loops at infinity in the mapping class group
(with
Aaron Abrams,
Noel Brady,
Pallavi Dani, and
Robert
Young), Michigan Math J., Volume 61, Issue 4 (2012), 867874.
(math.GR/1109.6048)
We study the Dehn function at infinity in the mapping class group,
finding a polynomial upper bound of degree four. This is the same upper bound that holds for arbitrary rightangled Artin groups.

 Length spectra and degeneration of flat metrics (with Chris Leininger and Kasra
Rafi), Inventiones Math., Vol 182, Issue 2 (2010), 231277.
(math.GT/0907.2082;
More recent)
We consider the class of flat metrics on surfaces (those induced by quadratic differentials)
with respect to the marked length spectrum of simple closed curves.
We show that a set of curves is "spectrally rigid" if and only if the set is dense in PMF.
In particular, knowing the lengths of all simple closed curves is enough to determine a flat metric, but no finite
set of curves suffices.
We construct an embedding of Flat(S) into the space of geodesic currents on S
and build a boundary for Flat(S) consisting of "mixed structures": part
flat metric, part foliation.

 Divergence of geodesics in Teichmüller space and the mapping class
group
(with Kasra Rafi),
GAFA, Volume 19, Issue 3 (2009), 722742.
(math.GT/0611.5359)
We show that both Teichmüller space (with the Teichmüller metric) and the mapping class group
(with a word metric) have geodesic divergence that is intermediate between the linear rate of
flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in
Teichmüller space, we find that their divergence is at most quadratic. Furthermore, this
estimate is shown to be sharp via examples of pairs of rays with exactly quadratic divergence.
The same statements are true for geodesic rays in the mapping class group. We explicitly describe
efficient paths "near infinity" in both spaces.

 Stars at infinity in Teichmüller space in deep freeze.
We study Karlsson's conjecture relating the intersection patterns of metric halfspaces in Teichmüller space
to the intersection numbers of foliations in PMF.

 Curvature, stretchiness, and dynamics
In the Tradition of Ahlfors and Bers IV,
Contemp. Math. 432 (2007).
Here, we present the definition of stretchiness, a fourpoint curvature condition used in a
previous work for the study of Teichmüller space, and elaborate some of its properties and
applications. We provide computations in some Hilbert geometries and in certain Banach spaces to
illustrate that stretchiness is wellsuited to the study of nonRiemannian metric spaces.

 Thin triangles and a multiplicative ergodic theorem for
Teichmüller geometry
University of Chicago dissertation (2005).
(math.GT/0508046)
We show that the Teichmüller metric satisfies a fourpoint curvature condition which provides a
version of largescale negative curvature. Using this, we derive a multiplicative ergodic theorem,
or "ray approximation," for the random action of the mapping class group: we show that almost
every sample path is tracked sublinearly by a geodesic.

Other research activity.
I am running a research team called the Metric Geometry and Gerrymandering Group (MGGG).
We maintain a website for our planned Geometry
of Redistricting Summer School, which includes information
for signing up for our mailing list.
I'll be hosting a 10person Research cluster
on Rational Billiards at Tufts in Summer 2017. There was a similar cluster in Summer 2014
on the topic of Random Groups.
I ran the Undergraduate Faculty Program
at PCMI 2012.
This was a Research Lab in
geometric group theory for 16 college professors.
Principal investigator for NSF grant DMS0906086, Metric geometry of groups and surfaces,
20092012, DMS1207106, Finer coarse geometry,
20122015, and DMS1255442, CAREER: Finer coarse geometry,
20132018.
I was a member of a recent
SQuaRE (small
working group) at the American Institute of Mathematics (AIM)
on Higher divergence functions together with
Aaron Abrams, Noel Brady, Pallavi Dani, and Robert Young.
For the 200607 school year, I led a VIGREfunded Research Focus
Group at UC Davis on
the subject of Geometric Group Theory.
113 BromfieldPearson Hall  Tufts University  Medford, MA 02155
Moon.Duchin@tufts.edu
