Geometric Group Theory
Reading Group
Our first meeting will be on Thursday, 02/08.
Time: 2:00-3:00pm on Thursdays
Location: 3306 at CUNY Graduate Center
Organizers:
Yushan Jiang (yjiang2@gradcenter.cuny.edu)
Weiyan Lin (wlin@gradcenter.cuny.edu)
Zhihao Mu (zmu@gradcenter.cuny.edu)
Jason Behrstock (jason.behrstock@lehman.cuny.edu)
Useful Links:
Spring 2024
February 8
Speaker: Weiyan Lin
Title: On hyperbolicity of free splitting and free factor complexes I
February 15
No meeting (conflict with Dennis' class)
February 22
No meeting (Monday schedule)
February 29
Speaker: Junjie Chen
Title: Construction of derived functor and its application to tensor product
March 7
Speaker: Weiyan Lin
Title: On hyperbolicity of free splitting and free factor complexes II
March 14
Speaker: Yushan Jiang
Title: Quasi-isometric rigidity of H^n (n>2) and Tukia’s theorem from complex analysis I
March 21
Speaker: Yushan Jiang
Title: Quasi-isometric rigidity of H^n (n>2) and Tukia’s theorem from complex analysis II
April 19
Speaker: Yushan Jiang
Title:
Fall 2023
We plan to read lecture notes from a course by Alex Wright about Teichmuller theory, CAT(0) cube complex and hierarchically hyperbolicity over this semester. But the topics are not restricted to those, please let us know if you have anything that wants to discuss in the reading group.
Here is a good survey for hierarchically hyperbolic spaces/groups by Alessandro Sisto.
For more resources about HHS: comet.lehman.cuny.edu/behrstock/HHS.html
September 21
Speaker: Zhihao Mu
Title: Geodices guessing lemma
Abstract: We will introduce the geodices guessing lemma(section 4,12,13 in Alex's notes). It is a fundamental tool to prove the hyperbolicity of the curve graph of surfaces.
September 28
Speaker: Yushan Jiang
Title: Hyperbolicity of Curve graph
Abstract: We will prove that the curve graph is Gromov-hyperbolic by using geodesic guessing and bicorn interpolations.
October 5
Speaker: Zhihao Mu
Title: Subsurface projections and bounded geodesic image theorem
October 12
Speaker: Yushan Jiang
Title: Teichmüller space and the systole map
October 19
Speaker: Yushan Jiang
Title: Coned-off Teichumüller space is quasi isometric to the curve graph
October 26
Speaker: Zhihao Mu
Title: Consistent conditions and bounded geodesic images
November 2
Speaker: Weiyan Lin
Title: The Farey Graph
November 9
No meeting
Spring 2023
The goal of this reading group is for graduate students to present some research topics in geometric group theory. Topics for this semester include right-angled Artin groups, mapping class groups, quasi-flats theorem, geodesic currents and more. If you have other topics that you wish to present, feel free to email one of the organizers.
February 2
Organization Meeting
February 9
Speaker: Yushan Jiang
Title: The Infinity, Hyperbolicity and Morse Lemma I
Abstract:
In this talk, I’ll recall some properties of real hyperbolic spaces and define what is a Gromov hyperbolic (or delta-hyperbolic) space. Then after introducing the Morse Lemma of (Gromov) hyperbolic space, I’ll illustrate how we extend quasi-isometry to the boundary at infinity and why hyperbolicity is a quasi-isometric invariant.
February 16
Speaker: Yushan Jiang
Title: The Infinity, Hyperbolicity and Morse Lemma II
Abstract:
February 23
Speaker: Zhihao Mu
Title: Coarse geometry of the mapping class group I
Abstract:
In this talk, we will study mapping class groups of finite type surfaces, which can be generated by finitely many ‘Dehn twists’. We will start from the definition of mapping class groups and explain the Nielsen-Thurston's classification theorem. One of the great tool to study mapping class groups is the curve complex, introduced by Harvey. We will discuss some of its properties and the relation between the mapping class group. A great reference for curve complexes is the note written by Saul Schleimer.
March 2
Speaker: Zhihao Mu
Title: Coarse geometry of the mapping class group II
Abstract:
In this hour, we will study the subsurface projections which induce coarsely well-defined maps between curve complexes of subsurfaces. Then we will introduce the Behrstock’s inequality which can be used to estimate the distance between two mapping class group elements in the Cayley graph.
March 9
No meeting
March 16
Speaker: Carol Badre
Title: Analogues and connections between geometric group theory and low-dimensional topology
Abstract:
There is an extensive connection between geometric group theory and low-dimensional topology. In this talk, I will discuss analogous theorems between the two areas- including Knesner's theorem, Grushko's theorem, Papakyriakopoulos' sphere theorem and Stalling's theorem of groups with more than one end.
March 23
Speaker: Yassin Chandran
Title: An introduction to infinite type surfaces
March 30
No meeting(UC Riverside workshop)
April 07
No meeting(Spring break)
April 13
No meeting(Spring break)
April 20
Speaker: Weiyan Lin
Title: (projectivized) geodesic currents, length function, and intersection form: constructions and properties
Abstract:
Bonahon’s foundational work on geodesic currents play an important role in understanding the geometry of the (compactified) Teichmuller space, and the dynamical properties of mapping class group and its elements. Moreover, a central point of Bonahon’s work on geodesic currents is the construction of the intersection form. Here in the context of free group of finite rank, we also have an analogous construction of geodesic currents. Moreover, as an analogy of the Teichmuller space, Culler and Vogtmann constructs the (projectivized) outer space, and it has been well studied ever since. Ultimately, there is a “natural” construction of the intersection form that arises in free groups. Throughout the talk, I will demonstrate the construction of these geometric objects, and present some important properties related to these objects.
April 27
Speaker: Weiyan Lin
Title: (projectivized) geodesic currents, length function, and intersection form: constructions and properties
Abstract: