Geometry/Topology Seminar

Fall 2013

Thursdays (and sometimes Tuesdays) 3-4pm, in Eckhart 308

- Tuesday September 17 at 1-2pm 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 one-dimensional 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 3-4pm 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 2-3pm in Eck 202
- Dave Witte Morris, University of Lethbridge, Canada
- Introduction to Bruhat-Tits buildings (1)
- Abstract: This minicourse will present a very brief introduction to Bruhat-Tits 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 torsion-free discrete subgroup of SL₂(Q_p) is free. In general, these buildings are metric spaces of non-positive curvature that provide p-adic analogues of Riemannian symmetric spaces.
- Wednesday October 9 at 3-4pm 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 3-4pm in Eck 308
- Martin Scharlemann, UCSB
- The Schonflies Conjecture and its spin-offs
- 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 2-3pm in Eck 202
- Dave Witte Morris, University of Lethbridge, Canada
- Introduction to Bruhat-Tits buildings (2)
- Abstract: This minicourse will present a very brief introduction to Bruhat-Tits 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 torsion-free discrete subgroup of SL₂(Q_p) is free. In general, these buildings are metric spaces of non-positive curvature that provide p-adic analogues of Riemannian symmetric spaces.
- Monday October 14 at 2-3pm in Eck 202
- Dave Witte Morris, University of Lethbridge, Canada
- Introduction to Bruhat-Tits buildings (3)
- Abstract: This minicourse will present a very brief introduction to Bruhat-Tits 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 torsion-free discrete subgroup of SL₂(Q_p) is free. In general, these buildings are metric spaces of non-positive curvature that provide p-adic analogues of Riemannian symmetric spaces.
- Thursday October 17 at 2-3pm 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 non-combinatorial). 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 Kirby-Siebenmann on combinatorial triangulations (PL structures), the existence of non-combinatorial triangulations by Edwards (and Cannon), the theory of non-combinatorial triangulations by Galewski-Stern and Matumoto and the reduction to a problem about homology 3-spheres with Rohlin invariant one which was solved by Manolescu.
- Thursday October 17 at 3-4pm in Eck 308
- Jenya Sapir, Stanford University
- Counting Non-Simple Closed Geodesics on Surfaces
- Abstract: We get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has given asymptotics for the growth of simple closed curves and curves with one self-intersection (respectively) with respect to length. No asymptotics for arbitrary self-intersection number are currently known, but we give coarse bounds for arbitrary self-intersection 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 3-4pm in Eck 308
- Ser-Wei 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 Duchin-Leininger-Rafi.
- Thursday October 31 at 2-3pm 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 2k-regular 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 non-primitive 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 3-4pm in Eck 308
- Tom Church, Stanford University
- Uniformly bounded generators for the Johnson filtration
- Abstract: It's well-known 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 = SL₂(Z), so SL_n(Z) is generated by elements that are "supported" on some SL₂(Z) subgroup. Similarly, the mapping class group Mod_g is generated by Dehn twists supported on a genus-1 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 genus-1 subsurfaces. However, Birman-Powell proved that the Torelli group IS generated by elements supported on genus-2 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 genus-G_k subsurfaces. Joint work with Andrew Putman.
- Thursday November 7 at 2-3pm 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 3-4pm 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 co-compact hyperbolic n-manifolds, 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 co-compact group. The purpose of the lecture is to discuss rigidity properties of such invariant.
- Thursday November 14 at 3-4pm 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 Culler-Vogtmann outer space CV_n. Each of these groups also have elements acting in some sense hyperbolically (pseudo-Anosov 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 pseudo-Anosov 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 2-3pm in Eck 202
- Rohit Nagpal, UW-Madison
- Nakaoka's theorem and mod-p 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 FI-module 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 mod-p cohomology of the unordered configuration space C_n(M) of a compact manifold M.
- Thursday November 21 at 3-4pm 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 bi-Lipschitz 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 3-4pm 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 co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have C¹ 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 C¹ boundary is projectively equivalent to the unit ball. The proof uses tools from dynamics, geometric group theory, and algebraic groups.
- Monday January 13 at 3-4pm 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 R-trees (Culler, Morgan, Levitt, Lustig, and others).
- Monday January 13 at 3-4pm 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 R-trees (Culler, Morgan, Levitt, Lustig, and others).
- Monday January 13 at 3-4pm 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 R-trees (Culler, Morgan, Levitt, Lustig, and others).
- Thursday February 27 at 3-4pm 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 2-dimensional subcomplexes. Joint work with A. Costa.

For questions, contact

Benson Farb, farb at math.uchicago.edu
Alex Eskin, eskin at math.uchicago.edu
Shmuel Weinberger, shmuel at math.uchicago.edu
Howard Masur, masur at math.uchicago.edu
Danny Calegari, dannyc at math.uchicago.edu
Qian Yin, qyin at math.uchicago.edu
Sebastian Hensel, hensel at math.uchicago.edu
Jonathan Chaika, jonchaika at math.uchicago.edu