Thursday, October 22, 2015 at 3pm in Bolz Hall (BO) 434
Valentina Disarlo, Indiana University
On the geometry of the flip graph
Abstract
The flip graph of an orientable punctured surface is the graph whose vertices are the ideal triangulations of the surface (up to isotopy) and whose edges correspond to flips. Its combinatorics is crucial in works of Thurston and Penner’s decorated Teichmuller theory. In this talk we will explore some geometric properties of this graph, in particular we will see that it provides a coarse model of the mapping class group in which the mapping class groups of some subsurfaces are strongly convex. We will also establish some bounds on the growth of the diameter of the flip graph modulo the mapping class group, extending a result of Sleator-Tarjan-Thurston. This is a joint work with Hugo Parlier.
Thursday, October 29, 2015 at 1:50pm in Enarson Classroom Building (EC) 240
Nick Salter, University of Chicago
4-manifolds can be surface bundles over surfaces in many ways
Abstract
An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the situation is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is "finitely many"? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on an asymptotic version of the "how many?" question. Time permitting, we will discuss some connections with the homology of the Torelli group, (non)-realization problems a la Nielsen and Morita, and symplectic topology.
Thursday, November 12, 2015 at 1:50pm in Cockins Hall (CH) 240
Jeff Meyer, University of Oklahoma
Geodesics & Surfaces: A Rigid Interaction
Abstract
In this talk, I will discuss recent joint work with Benjamin Linowitz and Paul Pollack on the interplay between closed geodesics and totally geodesic surfaces in arithmetic hyperbolic 3-manifolds. It has been shown that the commensurability class of an arithmetic hyperbolic 3-manifold is completely determined by the set of lengths of closed geodesics (called the length set) as well as by the set of totally geodesic surfaces. It is natural to ask whether the commensurability class is determined by only the lengths of closed geodesics lying on totally geodesic surfaces. We answer this question and go on to quantitatively understand (using counting techniques from analytic number theory) how this “totally geodesic length set” lies within the "full length set." To explain these results, I will give a gentle introduction to the theory of arithmetic hyperbolic 3-manifolds via the concrete example of the figure-eight knot complement, and I will go on to explain how manifolds, their surfaces, and their geodesics relate to division algebras, their subalgebras, and their maximal subfields.
Thursday, December 3, 2015 at 1:50pm in Cockins Hall (CH) 240
Wouter van Limbeek, University of Michigan
Rigidity of convex divisible domains in flag manifolds
Abstract
A projective structure on a manifold is a local modeling of the geometry on the geometry of projective space. Projective structures are usually lack rigidity: E.g. any hyperbolic manifold is canonically projective, but oftentimes the structure can be deformed. There are also projective structures on other manifolds altogether. A natural generalization of these structures is obtained by modeling the local geometry on other Grassmannians. In contrast to the plethora of examples of projective structures, we establish rigidity in this new context: We prove that in the Grassmannian of p-planes in R2p, p>1, every bounded convex domain with a compact quotient is a symmetric space. This is joint work with Andrew Zimmer.
Friday, December 11, 2015 at 11:30am in Baker Systems Engineering (BE) 188
Yago Antolin-Pichel, Vanderbilt University
The Dehn fillings theorem and applications
Abstract
In this talk I will recall the version of the Dehn fillings theorem proved by Dahmani, Guirardel and Osin. I will present an extension of this theorem and use it to show that groups hyperbolic relative to residually finite groups satisfying the Farrell-Jones conjecture, also satisfy this conjecture. This is based on a joint work with R.Coulon and G.Gandini.
Thursday, January 14, 2016 at 1:50pm in Cockins Hall (CH) 240
Elizabeth Fink, University of Ottawa
Morse geodesics in lacunary hyperbolic groups
Abstract
A geodesic is Morse if quasi-geodesics connecting points on it stay uniformly close. Such geodesics mark hyperbolic directions in the Cayley graph of a group. I will use combinatorial tools to study the geometry of lacunary hyperbolic graded small cancellation groups and show that they contain Morse geodesics. Further I will outline in a simple example an explicit but longer way to find Morse geodesics in such groups. This is joint work with R. Tessera.
Thursday, February 4, 2016 at 1:50pm in Cockins Hall (CH) 240
Patrick Reynolds, Miami University
Boundaries of some hyperbolic Out(F)-graphs
Abstract
After giving some background, we will give a description of the boundaries of two hyperbolic graphs on which the outer automorphism group of a free group admits an interesting action. This is joint work with M. Bestvina, and with M. Bestvina and M. Feighn.
Thursday, February 11, 2016 at 1:50pm in Cockins Hall (CH) 240
Anton Lukyanenko, University of Michigan
Diophantine approximation in the Heisenberg group
Abstract
The Heisenberg group arises both as a simple example of a nilpotent Lie group, and as the boundary of complex hyperbolic space. Studying it from the geometry-of-numbers perspective, we ask how well a generic point can be approximated by a rational point. Surprisingly, we obtain two natural ways make the question precise. The resulting Carnot Diophantine approximation applies to a broader class of nilpotent groups, while Siegel Diophantine approximation is directly related to complex hyperbolic geometry. This is joint work with Joseph Vandehey.
Postponed
Priyam Patel, Purdue University
Lifting curves simply in finite covers
Abstract
It is a well known result of Peter Scott that the fundamental groups of surfaces are subgroup separable. This algebraic property of surface groups also has important topological implications. One such implication is that every immersed (self-intersecting) closed curve on a surface lifts to an embedded (simple) one in a finite cover of the surface. A natural question that arises is: what is the minimal degree of a cover necessary to guarantee that a given closed curve lifts to be embedded? In this talk we will discuss various results answering the above question for hyperbolic surfaces, as well as several related questions regarding the relationship between geodesic length and geometric self-intersection number. Some of the work that will be presented is joint with T. Aougab, J. Gaster, and J. Sapir.
Thursday, April 21, 2016 at 1:50pm in Cockins Hall (CH) 240
Jingyin Huang, McGill University
Groups quasi-isometric to right-angled Artin groups
Abstract
We are motivated by the following question: suppose a finitely generated group H is quasi-isometric to a right-angled Artin group G, what kind of rigidity properties should we expect for H? In this talk, I will show that if the outer automorphism group of G is finite, then H admits a proper and cocompact action on a CAT(0) cube complex, which has an equivariant "fibration" over the right-angled building associated with G. If time allows, I will also discuss how does this leads to commensurability results in several cases. This is joint work with Bruce Kleiner. No previous knowledge about right-angled Artin groups and right-angled buildings is required.
Thursday, April 28, 2016 at 1:50pm in Cockins Hall (CH) 240
Tullia Dymarz, University of Wisconsin
Day's fixed point theorem, group cohomology and quasi-isometric rigidity
Abstract
To prove that a finitely generated group G that is quasi-isometric to hyperbolic n-space (for n>2) is virtually a lattice in the isometry group, Cannon-Cooper first construct a representation of G into the group of quasi-conformal maps of the boundary (n-1)-sphere. Then they appeal to Tukia's theorem on quasiconformal maps to conjugate the image of G into the the conformal group to finish the proof. We show how to adapt these ideas to prove similar results on groups quasi-isometric to certain solvable Lie groups. In particular we will talk about recent joint work with Xie that extends Tukia's theorem to boundaries of certain negatively curved homogeneous spaces.