Geometry/Topology Seminar
Fall 2013
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Tuesday September 17 at 12pm in Eck 308
 Peter Lee, Toronto
 The Pure Virtual Braid Groups and Representation Stability

Abstract: The virtual braid group is an extension of
the classical braid group by the symmetric group, introduced
by L. Kauffman, and forms a natural setting in which to
explore the concept of representation stability proposed by
T. Church and B. Farb. We will review the definition of the
virtual braid group vB_{n} and its
subgroup PvB_{n} of pure virtual braids,
the relation of these groups to the classical braid groups
and the groups of string motions, and the action of the
symmetric group S_{n} on the cohomology
modules H^{k}(PvB_{n},Q). We then
give a description of
H^{k}(PvB_{n},Q) as sums of
S_{n}modules induced from certain
onedimensional representations of specific subgroups of
S_{n}. We will recall Church and Farb's
notion of representation stability and show how the above
description allows us to conclude that
H^{k}(PvB_{n},Q) is uniformly
representation stable. Time permitting, we will discuss what
is known about the decomposition of
H^{k}(PvB_{n},Q) into irreducible
S_{n} modules, including the multiplicity
of the trivial and alternating representations, and some of
the techniques available to uncover this decomposition.

 Thursday October 3 at 34pm in Eck 308
 Andy Putman, Rice University
 On the second homology group of the Torelli subgroup of
Aut(F_{n})

Abstract: We prove that the second homology group of
the Torelli subgroup of Aut(F_{n}) is
finitely generated as a GL(n,Z)module. This is joint work
with Matthew Day.

 Wednesday October 9 at 23pm in Eck 202
 Dave Witte Morris, University of Lethbridge, Canada
 Introduction to BruhatTits buildings (1)

Abstract: This minicourse will present a very brief
introduction to BruhatTits buildings, and describe several
applications in group theory that are of geometric or
topological interest. The simplest buildings are trees, and
can be used to prove Ihara's Theorem that every torsionfree
discrete subgroup of
SL_{2}(Q_{p}) is free. In
general, these buildings are metric spaces of nonpositive
curvature that provide padic analogues of Riemannian
symmetric spaces.

 Wednesday October 9 at 34pm in Eck 203
 Martin Scharlemann, UCSB
 An overview of the Schonflies Conjecture

Abstract: This is meant to be a fairly accessible
talk, reviewing the antecedents of the Schonflies Conjecture
and its resolution in all dimensions other than 4. We
discuss why the conjecture is important, and the classic
approach to its resolution that has spawned much beautifully
pictorial mathematics, without actually succeeding.

 Thursday October 10 at 34pm in Eck 308
 Martin Scharlemann, UCSB
 The Schonflies Conjecture and its spinoffs

Abstract: A more technical talk, discussing several
methods that have been partially successful. An underlying
theme is that, although the conjecture has not yet been
settled, it interlocks with and has inspired much
interesting topology in dimensions three and four. Those
with only a casual interest or those who would just like to
get to the bottom line could skip this talk and instead
listen to the Emissaries of Misery sing about it at
http://www.math.ucsb.edu/~mgscharl/Vita.html

 Friday October 11 at 23pm in Eck 202
 Dave Witte Morris, University of Lethbridge, Canada
 Introduction to BruhatTits buildings (2)

Abstract: This minicourse will present a very brief
introduction to BruhatTits buildings, and describe several
applications in group theory that are of geometric or
topological interest. The simplest buildings are trees, and
can be used to prove Ihara's Theorem that every torsionfree
discrete subgroup of
SL_{2}(Q_{p}) is free. In
general, these buildings are metric spaces of nonpositive
curvature that provide padic analogues of Riemannian
symmetric spaces.

 Monday October 14 at 23pm in Eck 202
 Dave Witte Morris, University of Lethbridge, Canada
 Introduction to BruhatTits buildings (3)

Abstract: This minicourse will present a very brief
introduction to BruhatTits buildings, and describe several
applications in group theory that are of geometric or
topological interest. The simplest buildings are trees, and
can be used to prove Ihara's Theorem that every torsionfree
discrete subgroup of
SL_{2}(Q_{p}) is free. In
general, these buildings are metric spaces of nonpositive
curvature that provide padic analogues of Riemannian
symmetric spaces.

 Thursday October 17 at 23pm in Eck 207
 Rob Kirby, UC Berkeley
 Triangulations of manifolds, a history

Abstract: Recently, Ciprian Manolescu has put the
final piece in the puzzle of which manifolds have
triangulations (possibly noncombinatorial). I'll discuss
the history of this problem (but not Manolescu's work),
mentioning the early work in dimensions less than four, the
work of KirbySiebenmann on combinatorial triangulations (PL
structures), the existence of noncombinatorial
triangulations by Edwards (and Cannon), the theory of
noncombinatorial triangulations by GalewskiStern and
Matumoto and the reduction to a problem about homology
3spheres with Rohlin invariant one which was solved by
Manolescu.

 Thursday October 17 at 34pm in Eck 308
 Jenya Sapir, Stanford University
 Counting NonSimple Closed Geodesics on Surfaces

Abstract: We get coarse bounds on the number of
(nonsimple) closed geodesics on a surface, given upper
bounds on both length and selfintersection number. Recent
work by Mirzakhani and by Rivin has given asymptotics for
the growth of simple closed curves and curves with one
selfintersection (respectively) with respect to length. No
asymptotics for arbitrary selfintersection number are
currently known, but we give coarse bounds for arbitrary
selfintersection number and length. We show how to reduce
this problem to counting curves on a pair of pants, and give
explicit bounds with respect to both length and intersection
number in that case.

 Thursday October 24 at 34pm in Eck 308
 SerWei Fu, UIUC
 Length spectral rigidity for strata of Euclidean cone metrics

Abstract: When considering Euclidean cone metrics on
a surface induced by quadratic differentials, there is a
natural stratification by prescribing cone angles. I will
describe a simple method to reconstruct the metric locally
using the lengths of a finite set of closed curves, which
can be called a local length rigidity problem. However, the
main discussion will be on the surprising result that a
finite set of simple closed curves cannot be length
spectrally rigid when the stratum has enough complexity.
This is extending a result of DuchinLeiningerRafi.

 Thursday October 31 at 23pm in Eck 202
 Doron Puder, Einstein Institute of Mathematics (Hebrew University Jerusalem)
 Measure Preserving Words are Primitive

Abstract: We establish new characterizations of
primitive elements and free factors in free groups, which
are based on the distributions they induce on finite groups.
For every finite group G, a word w in
the free group on k generators induces a word map
from G^{k} to G. We say that w is measure
preserving with respect to G if given uniform
distribution on G^{k}, the image of this
word map distributes uniformly on G. It is easy
to see that primitive words (words which belong to some
basis of the free group) are measure preserving w.r.t. all
finite groups, and several authors have conjectured that the
two properties are, in fact, equivalent. In a joint work
with O. Parzanchevski we prove this conjecture. What we
really prove is the following. Take k independent
random permutations on n points (n
large). This defines a random 2kregular graph on
n points. What is the probability
P(w,n) that when you walk along some fixed w
starting at the vertex 1, you return to 1? It is easy to see
that when w is primitive, this probability is
exactly 1/n. We show that the probability for
nonprimitive w is strictly more than
1/n, and can be approximated using some algebraic
properties of w. This result, in turn, can be
used to obtain new results (and new proofs of known results)
regarding expansion of random graphs, and to prove that
random graphs are nearly optimal expanders.

 Thursday October 31 at 34pm in Eck 308
 Tom Church, Stanford University
 Uniformly bounded generators for the Johnson filtration

Abstract: It's wellknown that
SL_{n}(Z) is generated by elementary
matrices. The elementary matrices E_{ij}
and E_{ji} are contained in the subgroup
E_{ij}, E_{ji} =
SL_{2}(Z), so SL_{n}(Z) is
generated by elements that are "supported" on some
SL_{2}(Z) subgroup. Similarly, the
mapping class group Mod_{g} is generated
by Dehn twists supported on a genus1 subsurface. The same
question about subgroups is much harder! For congruence
subgroups SL_{n}(Z,p), asking whether
SL_{n}(Z,p) is generated by elementary
matrices is essentially equivalent to the Congruence
Subgroup Property. Johnson proved that the Torelli group
Mod_{g}[1] is NOT generated by elements
supported on genus1 subsurfaces. However, BirmanPowell
proved that the Torelli group IS generated by elements
supported on genus2 subsurfaces. I will give an overview of
such "generated by elements of bounded support" results, and
explain the ideas behind a new theorem: for every term
Mod_{g}[k] of the Johnson filtration,
there is a constant G_{k} so that
Mod_{g}[k] is generated by elements
supported on genusG_{k} subsurfaces.
Joint work with Andrew Putman.

 Thursday November 7 at 23pm in Eck 202
 Tsachik Gerlander, Einstein Institute of Mathematics (Hebrew University Jerusalem)
 Lattices in Amenable Groups

Abstract: Let G be a locally compact amenable group.
We discuss the question whether every closed subgroup of
finite covolume in G is cocompact. Joint work with U. Bader,
P.E. Caprace and S. Mozes.

 Thursday November 7 at 34pm in Eck 308
 Andres Sambarino, University of Chicago
 On entropy, regularity and rigidity

Abstract: Convex representations are a class of
representations of hyperbolic groups into
SL(d,R). This class contains convex cocompact
hyperbolic nmanifolds, convex projective structures on
closed manifolds, Hitchin representations of surface groups,
among others. The entropy of such representation is an
invariant, analogous to the Hausdorff dimension of the limit
set of a convex cocompact group. The purpose of the lecture
is to discuss rigidity properties of such invariant.

 Thursday November 14 at 34pm in Eck 308
 Catherine Pfaff, Marseille
 Lone Axes in Outer Space

Abstract: (Joint work with Lee Mosher) As with
SL(2,R) acting on hyperbolic space, a central
method for studying a mapping class group is to study its
action on its Teichmuller space and a central method for
studying an outer automorphism group of a free group
Out(F_{n}) is to study its action on its
CullerVogtmann outer space CV_{n}. Each
of these groups also have elements acting in some sense
hyperbolically (pseudoAnosov elements of mapping class
groups and fully irreducible outer automorphisms of free
groups). However, the analogy breaks down when one wants to
study the invariant axis for a fully irreducible. It appears
the correct object to study is actually a collection of
axes, an "axis bundle." By proving when the axis bundle for
a fully irreducible is just a single axis, we have
highlighted the setting where a fully irreducible also
behaves in this regard like a pseudoAnosov or hyperbolic
element. In fact, we have identified a setting where one can
actually quite easily identify when two fully irreducibles
are conjugate.

 Monday November 18 at 23pm in Eck 202
 Rohit Nagpal, UWMadison
 Nakaoka's theorem and modp cohomology of unordered configuration spaces

Abstract: Nakaoka showed in his 1960 paper that the
cohomology groups H^{t}(S_{n}, k)
stablizes when n is large enough. We cosider a
finitely generated FImodule V over k (a field of
characteristic p>0) and discuss generaliztions
of Nakaoka's result to the sequence of cohomology groups
H^{t}(S_{n}, V_{n}). We
also mention it's geometric implication to modp
cohomology of the unordered configuration space
C_{n}(M) of a compact manifold
M.

 Thursday November 21 at 34pm in Eck 308
 Dominic Dotterrer, University of Chicago
 Dilation of maps from complexes to space

Abstract: In the early 80's, Bourgain used random
graphs as an example of a metric space which required a
large biLipschitz constant to embed into Euclidean space,
i.e. distances in the graph cannot be commensurately
compared to distances in Euclidean space. We investigate a
cohomological analogue to higher dimensions. Specifically,
can the areas of simplicial minimal surfaces in a simplicial
complex be compared to areas of minimal surfaces in
Euclidean space? Just as Bourgain did, we use random
simplicial complexes to give examples which require a large
amount of dilation of areas to embed. In fact, the estimate
is even more dramatic than for graphs.

 Thursday December 5 at 34pm in Eck 308
 Andrew Zimmer, Michigan
 Rigidity of complex convex divisible sets

Abstract: An open convex set in real projective
space is called divisible if there exists a discrete group
of projective automorphisms which acts cocompactly. There
are many examples of such sets and a theorem of Benoist
implies that many of these examples are strictly convex,
have C^{1} boundary, and have word
hyperbolic dividing group. In this talk I will discuss a
notion of convexity in complex projective space and show
that every divisible complex convex set with
C^{1} boundary is projectively equivalent
to the unit ball. The proof uses tools from dynamics,
geometric group theory, and algebraic groups.

 Monday January 13 at 34pm in Eck 308
 Lee Mosher, Rutgers University
 The geometry, topology, and dynamics of free groups (I)

Abstract: Over a long span of time, a geometric
understanding of free groups has emerged from the study of
some very basic questions, such as: How do you tell when a
set of elements of a free group forms a basis up to
conjugacy (Whitehead's algorithm)? What are the possible
values of the cyclically reduced word norm growth for an
automorphism of F_{n}? What are the
possible ranks of fixed subgroups of automorphisms of
F_{n} (the Scott Conjecture, now a
theorem of Bestina and Handel)? Motivated by these questions
and others, in these lectures we shall survey the geometric
methods which over the last 25 years have led to a quite
revolution in the study of free groups. Our survey will
include various deformation spaces of geometric/topological
structures, such as the outer space of the free group
(introduced by Culler and Vogtmann), the sphere complex aka
the free splitting complex (Hatcher), and the free factor
complex (Hatcher and Vogtmann). We also survey the dynamical
theory of Out(F_{n}), the outer automorphism group,
which developed along the lines of Thurston's dynamical
theory for mapping class gorups, using tools of relative
train track maps (Bestvina and Handel), attracting
laminations (Bestvina, Feighn, and Handel), and Rtrees
(Culler, Morgan, Levitt, Lustig, and others).

 Monday January 13 at 34pm in Eck 308
 Lee Mosher, Rutgers University
 The geometry, topology, and dynamics of free groups (II)

Abstract: Over a long span of time, a geometric
understanding of free groups has emerged from the study of
some very basic questions, such as: How do you tell when a
set of elements of a free group forms a basis up to
conjugacy (Whitehead's algorithm)? What are the possible
values of the cyclically reduced word norm growth for an
automorphism of F_{n}? What are the
possible ranks of fixed subgroups of automorphisms of
F_{n} (the Scott Conjecture, now a
theorem of Bestina and Handel)? Motivated by these questions
and others, in these lectures we shall survey the geometric
methods which over the last 25 years have led to a quite
revolution in the study of free groups. Our survey will
include various deformation spaces of geometric/topological
structures, such as the outer space of the free group
(introduced by Culler and Vogtmann), the sphere complex aka
the free splitting complex (Hatcher), and the free factor
complex (Hatcher and Vogtmann). We also survey the dynamical
theory of Out(F_{n}), the outer automorphism group,
which developed along the lines of Thurston's dynamical
theory for mapping class gorups, using tools of relative
train track maps (Bestvina and Handel), attracting
laminations (Bestvina, Feighn, and Handel), and Rtrees
(Culler, Morgan, Levitt, Lustig, and others).

 Monday January 13 at 34pm in Eck 308
 Lee Mosher, Rutgers University
 The geometry, topology, and dynamics of free groups (III)

Abstract: Over a long span of time, a geometric
understanding of free groups has emerged from the study of
some very basic questions, such as: How do you tell when a
set of elements of a free group forms a basis up to
conjugacy (Whitehead's algorithm)? What are the possible
values of the cyclically reduced word norm growth for an
automorphism of F_{n}? What are the
possible ranks of fixed subgroups of automorphisms of
F_{n} (the Scott Conjecture, now a
theorem of Bestina and Handel)? Motivated by these questions
and others, in these lectures we shall survey the geometric
methods which over the last 25 years have led to a quite
revolution in the study of free groups. Our survey will
include various deformation spaces of geometric/topological
structures, such as the outer space of the free group
(introduced by Culler and Vogtmann), the sphere complex aka
the free splitting complex (Hatcher), and the free factor
complex (Hatcher and Vogtmann). We also survey the dynamical
theory of Out(F_{n}), the outer automorphism group,
which developed along the lines of Thurston's dynamical
theory for mapping class gorups, using tools of relative
train track maps (Bestvina and Handel), attracting
laminations (Bestvina, Feighn, and Handel), and Rtrees
(Culler, Morgan, Levitt, Lustig, and others).

 Thursday February 27 at 34pm in Eck 308
 Michael Farber, Warwick
 Fundamental groups of large random spaces

Abstract: I will discuss several probabilistic
models producing random simplicial complexes and will
describe properties of their fundamental groups (torsion and
cohomological dimension). Besides, I will examine the
probabilistic validity of the Whitehead conjecture for
aspherical 2dimensional subcomplexes. Joint work with A.
Costa.
For questions, contact