Our first meeting in Spring 2026 will be on Thursday, 2/19.
Time: 11:30-1:00pm on Thursdays
Location: 9116 at CUNY Graduate Center
Organizers:
Jason Behrstock (jason.behrstock@lehman.cuny.edu)
Shaivi Chavan(schavan@gradcenter.cuny.edu)
Yushan Jiang (yjiang2@gradcenter.cuny.edu)
Weiyan Lin (wlin@gradcenter.cuny.edu)
Zhihao Mu (zmu@gradcenter.cuny.edu)
Useful Links:
February 19
Speaker: Yushan Jiang
Title: Introduction to Pattern Rigidity
Abstract: Richard Schwartz studied quasi-isometric rigidity for rank-one lattices and discovered several striking phenomena. In particular, if a quasi-isometry of a rank-one symmetric space with dim>2 (for example, \mathbb{H}^n) coarsely permutes the cosets of certain special infinite-index subgroups of lattices—such as cusp groups or finitely many loxodromic cyclic subgroups—then this quasi-isometry must essentially be an isometry, in the sense that it is uniformly close to an actual isometry (equivalently, it induces a conformal map on the boundary at infinity).
Schwartz’s results also provide an alternative proof of the Mostow Rigidity Theorem, although in a more indirect way, since no group-equivariance assumptions are imposed.
In this talk, I will introduce these types of rigidity questions, explain Schwartz’s main results and strategies, and briefly discuss related work, such as that of Cashen-Macura on line patterns in non-abelian free groups. If time permits, I will also touch on the case of the hyperbolic plane, which involves more delicate analytic tools, including Patterson–Sullivan theory.
February 26
(Different time and location: 2:30-3:30pm at room 4214.03)
Speaker: Zhihao Mu
Title: Quasi-Hyperblic Planes in Hyperbolic Groups
Abstract: In this talk we will talk about Bonk-Kleiner's theorem, which shows that there is a quasi-isometrically embedded hyperbolic plane in hyperbolic groups that are not virtually free. This solves the geometric version of Gromov's surface subgroup problem (whether there is a surface subgroup in a one-ended hyperbolic group). The key ingredient is to study the metric properties of Gromov boundary. In the end, we will mention a related open question about curve graphs of finite-type surfaces.
March 12
Speaker: Eamon Olive
Title: Introduction to translation equivalence
Abstract: "I will give a basic introduction to Teichmüller space and Outer space showing how they are analogous concepts. Then we will construct a definition of hyperbolic equivalence of curves and use our analogy to introduce the concept of translation equivalence. Time permitting, we will discuss some open problems having to do with translation equivalence."
March 26
Speaker: Megha Bhat
Title: Large-scale geometry of homeomorphism groups
Abstract: When a group has a finite generating set, the associated word metric determines its large-scale geometry up to quasi-isometry. Rosendal extended this perspective to Polish groups with coarsely bounded generating sets, where boundedness depends on the topology of the group. Vlamis showed that certain transformation groups, including the identity component of homeomorphism groups of closed manifolds, admit generating sets that are bounded independently of the topology. In this talk, I will go over these results and also show that the same phenomenon holds for open tame manifolds.
April 23
Speaker: Shaivi Chavan
Title:
Abstract:
October 9
Speaker: Megha Bhat
Title: Curve Graphs I
Abstract: In this talk I will be introducing the curve graph and some of its basic properties. I will also talk about some results about curve graphs that have analogs in the world of fine curve graphs.
October 16
Speaker: Megha Bhat
Title: Curve Graphs II: The Fine Curve Graph
Abstract: In this talk I will define the fine curve graph and present a survey of results known so far. I will also present the common techniques used to study the fine curve graph and its Gromov boundary.
October 23
Speaker: Shaivi Chavan
Title: Asymptotic Cones, Relative Hyperbolicity, and Thickness
Abstract: What does a group look like when viewed from a distance? In this talk, we will explore how asymptotic cones can capture the large-scale geometry of spaces and groups by zooming out until all local details fade away. This perspective is closely related to one of the definitions of relative hyperbolicity, as outlined by Druţu and Sapir, where such groups exhibit an asymptotically tree-graded structure. If time allows, I will also discuss the concept of thickness, introduced by Behrstock, Druţu, and Mosher, which provides a stark contrast to relative hyperbolicity.
October 30
Speaker: Emma Hasson
Title: Braid groups
Abstract: In this talk we will define braids and the braid group along with some cool results about them. In particular, we will discuss methods for solving the word problem on braid groups and their computational complexities. This talk will be accessible without any required background.
November 6
Speaker: Junjie Chen
Title: quiver of affine monoid over F_q
Abstract: Let M=Aff(n,q) be the monoid of all affine map over a finite field F_q. \mathbb{C}M is a monoid ring. The quiver, Q(\mathbb{C}M) is a directed graph with vertexs the isomphism class of simple \mathbb{C}M module and number of arrow from simple module A to B equal dim(Ext^1(A,B)). In the talk, we will classify the simple \mathbb{C}M module by Mackey's little group method and compute the complete quiver of it by mackey decomposition and some other methods.
November 20
Speaker: Eamon Olive
Title: Abstract regular polytopes and applications of GGT
Abstract: We will introduce the basic theory of abstract regular polytopes (ARPs) and geometric regular polytopes, and we will show how we can apply classical results from geometric group theory to find novel results in the area and make progress towards open problems. This talk will require basic knowledge in groups and linear algebra but will not suppose any prior knowledge in ARPs or GGT.
March 13
Speaker: Yushan Jiang
Title: Patterson-Sullivan Theory, Regularity of Maps, and Mostow’s Rigidity
Abstract: We follow Sullivan’s argument, use dynamics to study the regularity of conjunctions between conformal group actions, and prove Mostow Rigidity Theorem. If time permits, we will also mention unexpected Tukia’s rigidity about differentiability.
March 20
Speaker: Yushan Jiang
Title: Patterson-Sullivan Theory, Regularity of Maps, and Mostow’s Rigidity II
Abstract: We continue last week’s arguments: start with two conformal group actions and a conjugation between them, we explain why there is a invariant ergodic measure and recall how to prove the Mostow Rigidity by using this measure. Later, we mention how to deal with some general cases (especially the infinite volume case) by introducing the critical exponents related to the geometry of orbits inside the metric space and the Patterson-Sullivan conformal measures which support on the outside: the boundary at infinity.
May 8
Speaker: Brett Berger (Stevens Institute of Technology)
Title: Regular Bi-Interpretations and the Fundamental Theorems of Geometry
Abstract: G. von Staudt in 1847 proved the Fundamental Theorem of Projective Geometry which states that for any vector space V of dimension at least 3 over any field F, the automorphism group of the lattice P(V) of linear subspaces under inclusion is exactly the set of those automorphisms induced by semilinear maps on V. There are similar results for other types of geometry, including affine, hyperbolic, spherical, and symplectic. In one direction, it is obvious that automorphisms of V induce automorphisms on P(V) since P(V) may be precisely defined in terms of V, which in turn may be defined in terms of F. In joint work with my advisor Alexei Miasnikov, we use a model theoretic tool known as interpretations to show a converse, that is a precise description of F in terms of P(V) which yields a regular bi-interpretation of the two. This is stronger than the above theorem and is an example of a bi-interpretation lying behind a theorem about some kind of rigidity. I’ll talk about the original proofs of the Fundamental Theorems, give an introduction to interpretations, and then discuss the above result.
September 26
Speaker: Yushan Jiang
Title: Introduction to the coarse geometry of finitely generated groups and its relationship to other subjects
October 3
Speaker: No meeting
October 10
Speaker: Zhihao Mu
Title: Simple random walks on Z^d(flats) and T_d(tree)
October 17
Speaker: Yushan Jiang
Title: Green's function and spectral radius
October 24
Speaker: Siobhan O'Connor
Title: Orbit-blocking words and potential positivity in F_2
Abstract:
We will answer two questions about automorphisms in F_2. First, given a word u in F_2 is there a word g such that g does not appear as a subword of any cyclically reduced automorphic image of u? Second, which words in F_2 are automorphic images of positive words (that is, words where all of the exponents are positive)?
October 31
Speaker: Weiyan Lin
Title: More on random walks
November 7
Speaker: Zhihao Mu
Title: Splittings of hyperbolic groups over two-ended groups
Abstract: A theorem of Bowditch states that a one-ended hyperbolic group which is not a triangle group splits over a two-ended group if and only if its boundary has a local cut point. As a corollary, spliting over virtually Z is a quasi-isometric invariant for hyperbolic groups. In this talk, we will give an elementary proof of this theorem due to Panos Papasoglu. https://arxiv.org/pdf/math/0107115
November 14
Speaker: Yushan Jiang
Title: Quasi-isometric rigidity of rank-1 nonuniform lattice
Abstract: When we consider a finitely generated group, we can look at the algebraic and (coarsely) geometric properties of it. The celebrated Gromov’s theorem on groups of polynomial growth tells us there are very deep relationships between these two kinds of properties. In this talk, I will introduce quasi-isometric rigidity of groups which follows Gromov’s program: what can we figure out by observing its coarse geometry. In particular, I will focus on a very strong results obtained by R. Schwartz about rank 1 non-uniform lattices (e.g. fundamental groups of non-compact complete hyperbolic 3-manifolds with finite volume): quasi-isometry implies commensurability. I will also introduce a higher dimensional Morse Lemma: quasi-flat lemma, obtained by Schwartz.
November 21
Speaker: Shaivi Chavan
Title: QI classification of graph manifold groups
Abstract: Given a collection of groups, G. Gromov asked the fundamental question of identifying which groups are quasi-isometric to those in G (rigidity) and which groups are quasi-isometric to one another (classification). In this talk, we look at the answer to the classification problem for graph manifold groups. We will look at a theorem of Behrstock and Neumann that shows that the fundamental groups of any two closed irreducible non-geometric graph manifolds are quasi-isometric.
December 5
Speaker: Junjie Chen
Title: An introduction to general properties of linear representations of finite groups
Abstract: I will introduce general properties of linear representations of finite groups. Topics will include Schur's Lemma, semisimple property of G-module, character of a representation and inner product of the characters. Also, I will introduce Restrict and induced representation and method of little group (Macky's method).
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
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: