Sponsored by
Boston College
The GEAR Network
HMI
NSF
SFI
|
The 9th William
Rowan
Hamilton Geometry and
Topology Workshop
on
Geometry
and
Groups
after
Thurston
August 27-31, 2013
The Hamilton Mathematics
Institute,
Trinity College Dublin
Mini-Course
Groups
acting on Quasi-Trees
by Ken
Bromberg (University of Utah, Salt Lake)
August
27-28, 2013
Schedule of Mini-Course
All minicourse lectures will be held in the Synge Lecture Hall in the
Hamilton Building (see campus map).
Time
|
Speaker
|
Talk
|
Tuesday
|
|
9:30
-
10:30
|
Ken Bromberg (Utah)
|
Groups acting on
Quasi-Trees I
|
10:30 - 11:00
|
COFFEE & DISCUSSION
|
11:00
-
12:00
|
Ken
Bromberg
(Utah) |
Groups acting on
Quasi-Trees II
|
12:00
-
2:00
|
LUNCH |
2:00
-
3:00
|
Richard Webb (Warwick)
|
Tight geodesics in
the curve complex |
| 3:00 - 4:00 |
DISCUSSION
|
Wednesday
|
|
| 9:30
-
10:30
|
Ken
Bromberg
(Utah) |
Groups acting on
Quasi-Trees III
|
10.30 - 11:00
|
COFFEE &
DISCUSSION |
| 11:00
-
12:00 |
Ken
Bromberg
(Utah) |
Groups acting on
Quasi-Trees IV
|
| 12:00
-
2:00 |
LUNCH |
| 2:00
-
3:00 |
Robert
Tang
(Warwick)
|
On various
projection maps in the curve complex |
3:00 - 4:00
|
Tarik Aougab (Yale)
|
Optimal intersection
numbers in the curve graph |
Lecture
Series
on
Geometry
and
Groups
after
Thurston
August
29-31,
2013
Schedule of Talks
All talks will be held in the MacNeil Lecture Hall in the
Hamilton
Building, TCD (see campus map).
Time
|
Speaker
|
Talk
|
Thursday
|
|
9.00-9.45
|
Ian Agol (Berkeley)
|
Virtually special
cube complexes |
9.45-10.15
|
COFFEE
|
10:15-11:00
|
Mladen
Bestvina
(Utah)
|
Towards the large
scale geometry of Out(F_n) |
| 11:15-12:00 |
Jeff
Danciger (Austin) |
Margulis spacetimes
via the arc complex |
12:00-2:00
|
LUNCH |
2:00-2:45
|
Richard
Schwartz
(Brown) |
"Turning Squares" |
3:00-3:45
|
Jean-Pierre
Otal
(Toulouse)
|
Small eigenfunctions
on hyperbolic surfaces |
4:00-4:45
|
Richard Canary (Michigan) |
The pressure metric
for convex representations |
Friday
|
|
| 9:00-9:45
|
Dave
Gabai
(Princeton)
|
Topologies of the
Curve Complex |
| 9.45-10.15 |
COFFEE |
| 10:15-11:00 |
Alan
Reid
(Austin)
|
Recognizing Surface
groups and Kleinian groups by their finite quotients |
| 11:15-12:00 |
Karen
Vogtmann
(Cornell) |
An outer space for
right-angled Artin groups |
| 12:00-2:00 |
LUNCH |
| 2:00-2:45 |
Jeff
Brock
(Brown)
|
TBA
|
| 3:00-3:45 |
Genevieve
Walsh
(Tufts) |
The Big Dehn
surgery
graph and the link of S^3
|
| 4:00-4:45 |
Yair
Minsky
(Yale)
|
Thick-skinned
3-manifolds
|
Saturday
|
|
| 10:00-10:45 |
Allen
Hatcher
(Cornell) |
Configuration Spaces
of Branched Polymers |
10.45-11.15
|
COFFEE
|
| 11.15-12:00 |
Dylan
Thurston
(Indiana)
|
Conformal embeddings
of graphs |
12.00-2.00
|
LUNCH
|
2.00-2.45
|
Mary Rees
(Liverpool) |
The Resident's View
revisited |
3.00-3.45
|
Mahan
Mitra
(Vivekananda) |
Homotopical Height |
| END OF
WORKSHOP |
Abstracts for Minicourse
Tarik Aougab (Yale)
Title: Optimal intersection numbers in the curve graph
Abstract: Let $S_{g,p}$ be the orientable surface of genus $g$ with $p$
punctures, and let $W(S)= 3g+p-4>0$. Given a natural number $k$, we
demonstrate a lower bound on the geometric intersection number for any
pair of curves that are a distance of $k$ apart in the corresponding
curve graph, and which grows "almost polynomially" of degree $k-2$, in
the sense that it grows faster than $(W(S)/2)^{c(k-2)}$, for any $c \in
(0,1)$. We use this to prove that curve graphs are uniformly
hyperbolic, and that train track splitting sequences project to
$R$-quasigeodesics in the curve graph of any essential subsurface,
where $R= O(W(S)^{2}). Time permitting, we will demonstrate a
construction which shows that these lower bounds on intersection
numbers are asymptotically sharp in some suitable sense.
Robert Tang (Warwick)
Title: On various projection maps in the curve complex
Abstract: In my talk, I will discuss some naturally occuring
quasiconvex subsets of the curve complex. I will then describe some
combinatorial operations, analogous to subsurface projections, which
approximate nearest point projection maps to these subsets.
Richard Webb (Warwick)
Title: Tight geodesics in the curve complex
Abstract: Tight geodesics were introduced by Masur and Minsky to
develop their hierarchies of curve complexes - these are used in
solving Thurston's ending lamination conjecture, as well as having many
applications in mapping class groups. Masur and Minsky showed between
any pair of vertices in the curve complex there are only finitely many
tight geodesics. Bowditch used hyperbolic 3--manifolds to refine
further these finiteness properties, in particular showing that the
mapping class group acts acylindrically on the curve complex. We shall
give a constructive proof of Bowditch's result and some new
consequences. We deduce via work of Bell and Fujiwara that curve
complexes have asymptotic dimension bounded above by an exponential in
Euler characteristic. Therefore by work of Bestvina, Bromberg and
Fujiwara, mapping class groups have a computable upper bound on their
asymptotic dimension.
Abstracts for Lecture Series
Ian Agol (Berkeley)
Title: Virtually special cube complexes
Abstract: We'll discuss some of the ingredients of the proof that
cubulated hyperbolic groups are virtually special. In particular we'll
discuss proofs of Wise's malnormal special quotient theorem and
cubulating malnormal amalgams of Hsu and Wise. This is joint work with
Daniel Groves and Jason Manning.
Mladen Bestvina (Utah)
Title: Towards the large scale geometry of Out(F_n)
Abstract: I will review the Masur-Minsky theory, which describes the
large scale geometry of mapping class groups. Then I will talk about
the current progress towards an analogous theory for Out(F_n).
Richard Canary (Michigan)
Title: The pressure metric for convex representations
Abstract: Using the thermodynamics formalism, we introduce a notion of
intersection for convex Anosov representations, show analyticity
results for the intersection and the entropy, and rigidity results for
the intersection. We use the renormalized intersection to produce an
Out(G)-invariant Riemannian metric on the smooth
points of the deformation space of convex, irreducible
representations of a word hyperbolic group G into SL(n,R) whose Zariski
closure contains a generic element. In particular, we produce
mapping class group invariant Riemannian metrics on Hitchin
components which restrict to the Weil-Petersson metric on the Fuchsian
loci. Moreover, we produce Out(G)-invariant metrics on deformation
spaces of convex cocompact representations into PSL(2,C) and show that
the Hausdorff dimension of the limit set varies analytically over
analytic families of convex cocompact representations into any rank 1
semi-simple Lie group. This is joint work with Martin Bridgeman,
François Labourie and Andres Sambarino.
Jeff Danciger (Austin)
Title: Margulis spacetimes via the arc complex
Abstract: Margulis found the first examples of complete affine
three-manifolds with free fundamental group. Associated to each of
these manifolds, now called Margulis spacetimes, is a non-compact
hyperbolic surface and an infinitesimal deformation of that surface
which expands the lengths of closed geodesics in a controlled manner. I
will describe a basic construction, called infinitesimal strip
deformation, which generates all such expanding surface deformations in
an essentially unique way. Roughly, one inserts hyperbolic strips along
a collection of disjoint geodesic arcs which fill the surface. This is
joint work with François Guéritaud and Fanny Kassel.
Dave Gabai (Princeton)
Title:Topologies of the Curve Complex
Abstract: We compare the topology of the curve complex with its
classically defined simplicial structure and its subspace topology
induced from PML. For various surfaces, we show how to explicitly
construct essential spheres in these spaces.
Allen Hatcher (Cornell)
Title:Configuration Spaces of Branched Polymers
Abstract: This is an unfinished project with Bill Thurston from 2008,
with input also from Richard Kenyon. By a branched polymer we
mean a finite collection of closed unit disks in the plane such that
their union is connected and their interiors are disjoint. Let
BP(n) be the space of all such configurations with n disks. One
might guess that BP(n) has the same homotopy type as the space of all
configurations of n distinct points in the plane, an aspherical space
with fundamental group the braid group. However this is false for
n at least 8, by a simple argument that Bill found for the case n = 8
that generalizes to larger n as well, showing that BP(n) has a
nontrivial homotopy group in dimension 5 when n is 8 or more.
This depends strongly on restricting all the disks to have equal
radius. Dropping this requirement yields a larger configuration
space, where the sizes of all the disks can vary arbitrarily and
independently. It appears that this larger space does have the
same homotopy type as the space of point configurations, and an
argument for this will be sketched, after the argument for the
non-asphericity of BP(n) is explained. A weaker conjecture might
be that BP(n) at least has fundamental group the braid group, but we
were unable to decide this.
Mahan Mitra (Vivekananda)
Title: Homotopical Height
Abstract: We introduce the notion of homotopical height $ht_\CC(G)$ of
a finitely presented group $G$ within a class $\CC$ of smooth manifolds
with an extra structure (e.g. symplectic, contact, K\"ahler etc).
Homotopical height provides an obstruction to finding a $K(G,1)$ space
within the given class $\CC$. This leads to a hierarchy of these
classes in terms of "softness" or "hardness" a la Gromov. We show that
the classes of closed contact, CR, and almost complex manifolds as well
as the class of (open) Stein manifolds are soft. The classes $\SP$ and
$\CA$ of closed symplectic and complex manifolds exhibit intermediate
"softness" in the sense that every finitely presented group $G$ can be
realized as the fundamental group of a manifold in $\SP$ and a manifold
in $\CA$. For these classes, $ht_\CC(G)$ provides a numerical invariant
for finitely presented groups. We give explicit computations of these
invariants for some standard finitely presented groups. We use the
notion of homotopical height within the "hard" category of K\"ahler
groups to obtain partial answers to questions of Toledo
regarding second cohomology and second group cohomology of K\"ahler
groups. We also modify and generalize a construction due to Dimca,
Papadima and Suciu to give a large class of projective groups
(fundamental group of complex projective manifolds) violating property
FP. These provide counterexamples to a question of Koll\'ar. This is
joint work with Indranil Biswas, Dishant Pancholi
Jean Pierre Otal (Toulouse)
Title: Small eigenfunctions on hyperbolic surfaces
Abstract: The talk will discuss topological properties of Laplace
eigenfunctions on negatively curved surfaces, and in
particular the theorem that on any hyperbolic surface of genus g,
the number of Laplace eigenvalues below 1/4 is at most 2g-2.
Mary Rees (Liverpool)
Title: The Resident's View revisited
Abstract: It is in the nature of Bill Thurston's work that his
contributions to all of geometry, topology and dynamics have
become foundational in all three areas. It is impossible, for example,
to study dynamics in two dimensions without reference to Thurston's
isotopy classification of surface homeomorphisms. A related result
called ``Thurston's theorem'' is fundamental in complex dynamics. This
result is about postcritically finite branched coverings of the
two-sphere, up to the appropriate homotopy equivalence which is always
described as ``Thurston equivalence'', and can be viewed as an
analogue, for postcritically finite branched coverings, of the result
about surface homeomorphisms. The main technique in Thurston's proof
of both results is dynamical, an iteration on a Teichm\"uller
space -- and this sort of technique is also crucial in his
hyperbolisation theorems. I will talk about work on larger spaces
of branched coverings of the two-sphere, defined by relations on the
orbits of critical points, and, in particular, on the fundamental
groups of such spaces and the actions on the appropriate Teichm\"uller
space. This work has been ongoing since 1990 and the main publication
was in 2004 (Ast\'erisque 288), but the ideas promoted by Bill
Thurston, and our understanding of the geometry of the relevant
Teichm\"uller spaces, continue to develop, as I will try to describe.
Alan Reid (Austin)
Title: Recognizing Surface groups and Kleinian groups by their
finite quotients
Abstract: This talk will discuss recent work on distinguishing
certainclasses of finitely presented groups by their finite quotients.
If time permits this will include a surprising application of the work
of Agol and Wise.
Richard Schwartz (Brown)
Title: "Turning Squares"
Abstract: I'll describe a group action on the plane where the elements
act by infinite rectangle exchanges. The system is based on the idea of
turning the squares of a square grid by 90 degrees. These planar
systems have higher dimensional compactifications, and in the one case
I studied in detail the compactification has a renormalization scheme
and is related to the (2,4,infinity) triangle group and to the E4 Weyl
group. In my talk I'll show lots of computer pictures and try to
survey everything I know about these systems.
Dylan Thurston (Indiana)
Title: Conformal embeddings of graphs
Abstract: Given a branched topological covering $f: (S^2,P) \to (S^2,
P)$ of the sphere by itself, with branch values contained in $P$, can
$f$ be realised as a rational map? We give a positive criterion,
a counterpart to the obstruction William Thurston found in 1982.
We show that f is rational iff there is a metric spine $G$ for
$S^2\setminus P$ so that $f^{-n}(G)$ conformally embeds inside $G$ for
sufficiently large $n$. Here, a map $p: G \to H$ between metric graphs
is a \emph{conformal embedding} if, for almost all $y \in H$, \[
\sum_{f(x)=y} |f'(x)| <
1. \] (The intuition is that $G$ conformally embeds inside $H$ if a
slight thickening of $G$ conformally embeds as a Riemann surface inside
a slight thickening of $H$.)
This condition on conformal embeddings of graph relates, in turn, to
whether the corresponding virtual endomorphism of $\pi_1(S^2\setminus
P)$ is contracting in a non-standard metric on the group. This is
joint-work-in-progress with Kevin Pilgrim.
Karen
Vogtmann
(Cornell)
Title: An outer space for right-angled Artin groups
Abstract: The study of Outer space for a free group has been
largely driven by Thurston's study of surface mapping class groups via
their action on Teichmuller spaces. To extend these ideas to
rght-angled Artin groups (RAAGs) one needs a space which can
serve the same role, i.e. a contractible space with a proper action of
the outer automorphism group. I will discuss joint work with Ruth
Charney in which we construct such a space.
Genevieve Walsh (Tufts)
Title: The Big Dehn surgery graph and the link of S^3.
Abstract: A foundational result of Lickorish and Wallace allows one to
create a connected graph where the vertices correspond to closed
orientable 3-manifolds and the edges correspond to the existence of a
Dehn surgery. This graph was first considered by W. Thurston. We will
restate old results and questions in the context of this graph and give
some new results and questions related to this graph. For example, we
show that there are hyperbolic manifolds with weight one fundamental
group which are not obtained via surgery on a knot in S3.
We also explore the geometry of the graph and exhibit some interesting
subgraphs.
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