Columbia Geometric Topology Seminar

Spring 2020


Zoom link for Friday 5/1 (Chen):

Note: the meeting is password-protected; the password will be shared via the mailing list.


Organizer: Nick Salter.
The GT seminar meets on Fridays at 2:00pm Eastern time via Zoom. Check this page for the link to the Zoom meeting on Friday afternoons.

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Spring 2020

Date Speaker Title
January 24 Ryan Budney Isotopy in dimension 4
January 31 Nick Salter Framed mapping class groups and strata of abelian differentials
February 7 Jacob Russell Relative hyperbolicity in hierarchically hyperbolic spaces
February 14 Colin Adams Hyperbolicity and Turaev Hyperbolicity of Classical and Virtual Knots
February 21 Daniele Alessandrini Non commutative cluster coordinates for Higher Teichmüller Spaces
February 28 Abhijit Champanerkar Right-angled polyhedra and alternating links 
March 6 Peter Shalen Quantitative Mostow Rigidity
March 27 Keiko Kawamuro Detection of overtwisted contact structures, loose transverse links, and virtually loose transverse links
April 10 Harrison Bray Random walks on punctured convex real projective surfaces
April 24 Rylee Lyman Some new CAT(0) free-by-cyclic groups
May 1 Lei Chen Actions of Homeo and Diffeo groups on manifolds


Lei Chen, Caltech

May 1
: Actions of Homeo and Diffeo groups on manifolds

AbstractIn this talk, I discuss the general question of how to obstruct and construct group actions on manifolds. I will focus on large groups like Homeo(M) and Diff(M) about how they can act on another manifold N. The main result is an orbit classification theorem, which fully classifies possible orbits. I will also talk about some low dimensional applications and open questions. This is a joint work with Kathryn Mann.

Rylee Lyman, Tufts

April 24
Some new CAT(0) free-by-cyclic groups

Abstract:As with fundamental groups of 3-manifolds fibering over the circle, free-by-cyclic groups form a varied and interesting class of groups whose geometry depends in large part on the corresponding monodromy, in this case an outer automorphism of the free group. For example, Hagen and Wise showed that word-hyperbolic free-by-cyclic groups act virtually cospecially on CAT(0) cube complexes, while Gersten found an example of a free-by-cyclic group that cannot be even a subgroup of a CAT(0) group. Gersten's group admits a cyclic hierarchy, an iterated splitting as a graph of groups with free-by-cyclic vertex groups and cyclic edge groups, terminating in Z times Z. By contrast, we show that a large class of free-by-cyclic groups admitting an additional symmetry act geometrically on CAT(0) 2-complexes. Up to taking powers this includes mapping tori of all polynomially-growing palindromic and symmetric automorphisms. A key tool in the proof is our construction of so-called CTs for free products.

Harrison Bray, University of Michigan

April 10
: Random walks onn punctured convex real projective surfaces

Abstract I'll discuss the following result, joint with Giulio Tiozzo: Letting \Gamma be the representation of a punctured hyperbolic surface group in PSL(3,R) acting discretely, properly discontinuously, and with finite co-volume on a properly convex set in the projective plane, we have that hitting measure for a random walk on any \Gamma-orbit with certain moment conditions is singular with respect to the classical Patterson-Sullivan measure. The approach extends and adapts the strategy of Maher-Tiozzo for punctured hyperbolic surfaces. We also prove an essential global shadow lemma for finite volume convex real projective manifolds.


Keiko Kawamuro, University of Iowa

March 27
Detection of overtwisted contact structures, loose transverse links, and virtually loose transverse links

AbstractIn this talk, I study punctured abstract open books. Punctures give rise to a closed braid, that is identified with a transverse link in the supported contact structure via the Giroux correspondence. I will define a twist-left-veering open book and show that it supports an overtwisted contact structure. I will also show when the transverse links is loose (the compliment is overtwisted) and virtually loose. This is joint work with Tetsuya Ito.


Peter Shalen, UIC

March 6
: Quantitative Mostow Rigidity

Abstract: The Mostow Rigidity Theorem implies that the geometry of compact hyperbolic manifolds of dimension at least 3 is entirely determined by their topology. This means that any geometric invariant of a hyperbolic manifold may be regarded as a topological invariant. The theorem itself says nothing about the question of how these topological invariants are related to more classical topological invariants of a space. I will describe some progress on this question, which has been a focus of my research for many years. I will emphasize the connection between the volume of a hyperbolic 3-manifold M, which is its most important geometrically defined invariant, and the dimension of the vector space H_1(M,F_p) for certain primes p, which are its most classical computable topologically defined invariants. This includes joint work with Phil Wagreich, Marc Culler, Gilbert Baumslag, Ian Agol, and Jason DeBlois, and  some very recent work with Rosemary Guzman.

Abhijit Champanerkar, CUNY

February 28
Right-angled polyhedra and alternating links 

Abstract: To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link.  We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link. This is joing work with Ilya Kofman and Jessica Purcell. 

Daniele Alessandrini, Columbia

February 21
:Non commutative cluster coordinates for Higher Teichmüller Spaces

Abstract: In higher Teichmuller theory we study subsets of the character varieties

of surface groups that are higher rank analogs of Teichmuller spaces,
e.g. the Hitchin components and the spaces of maximal representations.

Fock-Goncharov generalized Thurston's shear coordinates and Penner's
Lambda-lengths to the Hitchin components, showing that they have a
beautiful structure of cluster variety.

Here we apply similar ideas to Maximal Representations and we find new
coordinates on these spaces that give them a structure of non-commutative
cluster varieties, in the sense defined by Berenstein-Rethak.

This is joint work with Guichard, Rogozinnikov and Wienhard.


Colin Adams, Williams College

February 14
Hyperbolicity and Turaev Hyperbolicity of Classical and Virtual Knots

Abstract: We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume.   We will discuss what is known about  this invariant. We further employ a construction of Turaev to associate a family of hyperbolic 3-manifolds of finite volume to any classical or virtual link, even if non-hyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic 3-manifolds associated via this Turaev construction.  We will talk about what is known.


Jacob Russell, CUNY

February 7
Title: Relative hyperbolicity in hierarchically hyperbolic spaces
Relative hyperbolicity and thickness describe incompatible ways that the non-negatively curved parts of a metric space can be organized. In several classes of spaces (Teichmuller space, Coxeter groups, 3-manifold groups) there exists a strict dichotomy between relative hyperbolicity and thickness that  produces strong geometric consequences. Behrstock, Drutu, and Mosher have thus asked for which additional classes of spaces can such a dichotomy be established. We investigate this question in the class of hierarchically hyperbolic spaces and produce a combinatorial criteria for detecting relative hyperbolicity. We apply this criteria to prove the separating curve graph of a surface has the relatively hyperbolic versus thick dichotomy.


Nick Salter, Columbia University

January 31
Title: Framed mapping class groups and strata of abelian differentials
Abstract: A holomorphic 1-form on a Riemann surface admits a geometric incarnation as a so-called translation surface. The moduli spaces of translation surfaces are known as strata. The dynamics of translation surfaces is an intense area of active study, but the topological properties of strata are almost entirely unknown. I will outline some work, joint with Aaron Calderon, aimed at obtaining information about the fundamental groups of strata by means of a monodromy representation into the mapping class group. The core of our approach is a study of the ``framed mapping class group'', a natural infinite-index subgroup that, very surprisingly, turns out to admit a very simple finite set of generators.


Ryan Budney, University of Victoria

January 24
Title: Isotopy in dimension 4
I will describe why the trivial knot S2-->S4 has non-unique spanning discs up to isotopy. This comes from a chain of deductions that include a description of the low-dimensional homotopy-groups of embeddings of S1 in S1xSn (for n>2), a group structure on the isotopy-classes of reducing discs of S1xDn, and the action of the diffeomorphism group Diff(S1xSn) on the embedding space Emb(S1, S1xSn).  Roughly speaking, these results say there is no direct translation from dimension 3 to 4, for the Hatcher-Ivanov theorems on spaces of incompressible surfaces. 

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