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: