Spring 2022 Schedule

Mar. 03 2022, Wenyuan Yang Peking University
Time: 15:00  17:00 Tencent room：86186617196 password: 123555 Title: Proper actions of 3manifold groups on finite product of quasitrees Abstract: Let M be a compact, connected, orientable 3manifold. In this talk, I will study when the fundamental group of M acts properly on a finite product of quasitrees. Our main result is that this is so exactly when M does not contain Sol and Nil geometries. In addition, if there is no $\widetilde{SL(2, \mathbb{R})}$ geometry either, then the orbital map is a quasiisometric embedding of $\pi_1(M)$. This is called property (QT) by BestvinaBrombergFujiwara, who established it for residually finite hyperbolic groups and mapping class groups. The main step of our proof is to show property (QT) for the classes of CrokeKleiner admissible groups and of relatively hyperbolic groups under natural assumptions. Accordingly, this yields that graph 3manifold and mixed 3manifold groups have property (QT). This represents joint work with N.T. Nguyen and S.Z. Han.

Mar. 10 2022, Martin Palmer Mathematical Institute of the Romanian Academy
Time: 15:0017:00 Zoom Id: 646 617 8889 password: 123456wu Title: Mapping class group representations via Heisenberg, Schrödinger and Stonevon Neumann Abstract: One of the first interesting representations of the braid groups is the Burau representation. It is the first of the family of Lawrence representations, defined topologically by viewing the braid group as the mapping class group of a punctured disc. Famously, the Burau representation is almost never faithful, but the k = 2 Lawrence representation is always faithful: this is a celebrated theorem of Bigelow and Krammer and it implies that braid groups are linear (embed into general linear groups over a field). Motivated by this, and by the open question of whether mapping class groups are linear, I will describe recent joint work with Christian Blanchet and Awais Shaukat in which we construct analogues of the Lawrence representations for mapping class groups of compact, orientable surfaces. Tools include twisted BorelMoore homology of configuration spaces, Schrödinger representations of discrete Heisenberg groups and the Stonevon Neumann theorem.

Mar. 17 2022, Xianchang Meng Shandong University
Time: 15:00  17:00 Tencent room：86186617196 password: 123555 Title: Distinct distances on hyperbolic surfaces Abstract: Erdős (1946) proposed the question of finding the minimal number of distinct distances among any N points in the plane. GuthKatz (2015) gave almost sharp answer for this question using incidence geometry and polynomial partitioning. We consider this problem in hyperbolic surfaces associated with cofinite Fuchsian groups, i.e. the volume of the surface is finite. We prove a lower bound of the same strength as GuthKatz. In particular, for any finite index subgroup of the modular group, we extract out the dependence of the implied constant on the index.

Mar. 24 2022, Edgar A. Bering IV Technion – Israel Institute of Technology
Time: 15:00  17:00 Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic Title: An exhaustion of the sphere complex by finite rigid sets Abstract: A subcomplex X < C of a simplicial complex is rigid if every locally injective, simplicial map X \to C arises as the restriction of an automorphism of $C$. Curve complexes and other surface complexes have been found to exhibit remarkable rigidity properties. Aramayona and Leininger proved that the curve complex of an orientable surface can be written as an increasing union of finite rigid sets. The sphere complex of a connect sum of $n$ copies of S^1 \times S^2 is an analog the curve complex of a surface used in the study of Out(F_n). In this talk I will present joint work with C. Leininger where we prove that there is an exhaustion of the sphere complex by finite rigid sets when n >= 3.

Mar. 31 2022，Bruno Martelli University of Pisa
Time: 15:00  17:00 Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic Title: Fibrations on higherdimensional hyperbolic manifolds Abstract: One of the most intriguing aspects in lowdimensional topology is the existence, discovered by Jorgensen in the late 70s, of hyperbolic 3manifolds that fiber over the circle. In this talk we will briefly review some aspects of this beautiful theory, with the notable contributions of Thurston, and more recently of Agol and Wise. Then we will show that this phenomenon is not restricted to dimension 3, by exhibiting some examples in dimension 4 and 5 (in even dimension, some critical points are necessary, and we talk about perfect circle valued Morse functions instead of fibrations). As a consequence, we will deduce that a finite type subgroup of a hyperbolic group needs not to be hyperbolic, thus answering a wellknown open question in geometric group theory (joint works with Battista, Italiano, and Migliorini)

Apr. 07 2022, John R. Parker Durham University
Time: 15:00  17:00
Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic
Title: Complex hyperbolic lattices
Abstract: I will give a survey on recent results about complex hyperbolic lattices. A lattice in a Lie group is a discrete subgroup whose quotient has finite Haar measure. In this talk I will concentrate on the Lie group SU(n,1), often restricting to the case where n=2. Elements of this group act as holomorphic isometries of complex hyperbolic space. Roughly speaking, there are four methods of constructing complex hyperbolic lattices. I will discuss how these are related, with particular emphasis on a family of nonarithmetic lattices.

Apr. 14 2022, Robert Tang, Xi’an JiaotongLiverpool University
Time: 15:00  17:00
Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic
Title: Largescale geometry of the saddle connection graph
Abstract: For a translation surface, the associated saddle connection graph has saddle connections as vertices, and edges connecting pairs of noncrossing saddle connections. This can be viewed as an induced subgraph of the arc graph of the surface. In this talk, I will discuss both the fine and coarse geometry of the saddle connection graph. We show that the isometry type is rigid: any isomorphism between two such graphs is induced by an affine diffeomorphism between the underlying translation surfaces. However, the situation is completely different when one considers the quasiisometry type: all saddle connection graphs form a single quasiisometry class. We will also discuss the Gromov boundary in terms of foliations. This is based on joint work with Valentina Disarlo, Huiping Pan, and Anja Randecker.

Apr. 21 2022，Weiyan Chen, Tsinghua University
Time: 15:00  17:00
Tencent room：86186617196 password: 123555
Title: Choosing points on cubic plane curves
Abstract: It is a classical topic to study structures of certain special points on smooth complex cubic plane curves, for example, the 9 flex points and the 27 sextactic points. We consider the following topological question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose n distinct points on every smooth cubic plane curve, for each given integer n? This work is joint with Ishan Banerjee.

Apr. 28 2022, Fangzhou Jin, Tongji University (Postponed)

May 05 2022, Brita Nucinkis Royal Holloway, University of London
Time: 15:00  17:00 Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic Title: Cohomological properties of Mackey functors for totally disconnected locally compact groups Abstract: Mackey functors for finite groups are well understood. In the early 2000s this was extended to infinite discrete groups, and their cohomological finiteness conditions have been expressed in terms of relative cohomology and finiteness conditions of classifying spaces for proper actions. In this talk I will indicate how one can extend the definition to totally disconnected groups, and will indicate some of the obstacles encountered here. This is ongoing work with Ilaria Castellano and Nadia Mazza.

May 12 2022 Inheok Choi, KAIST.
Time: 15:00  17:00 Zoom Meeting ID: 956 0945 4208 Passcode: hyperbolic Title: Typical behavior of random mapping classes and outer automorphisms Abstract: Random walks on spaces that manifest hyperbolic properties have been studied for decades, as a means to investigate the structure of the isometry groups of such spaces. Notable results in this topic include descriptions of generic random isometries and limit laws for displacements and translation lengths. A recent technique called the pivoting technique led to further elaborations in these directions. In this talk, I will describe properties of generic random isometries of Teichmuller space or Outer space. Namely, generic mapping classes are principal pseudoAnosovs that make displacements with many contracting subsegments. This frequently contracting property is shared with generic outer automorphisms, which are ageometric fully irreducibles and whose expansion factors differ from their inverses'. If time permits, I will explain how this description is related to (1) QingRafi's sublinearly Mosre boundaries and (2) limit laws for random 3manifolds.

May 19 2022 Hao Liang, Sun Yatsen University
Time: 15:0017:00 Tencent room：86186617196 password: 123555 Title: Homomorphisms to 3manifold groups Abstract: Following Sela's theory of limit groups (of free group), we define and study limit groups of compact 3manifold groups. We show that the family of compact 3manifold groups are equationally noetherian, which answers a question of Agol and Liu. The main application of our result is to give a positive answer to a question of Reid, Wang and Zhou about epimorphism sequence of closed orientable aspherical 3manifold groups. This is joint work with Daniel Groves and Michael Hull.

May 26 2022

Jun. 02 2022, Yi Jiang, Tsinghua University

Jun. 09 2022, Bin Sun, Oxford University
Fall 2021 Schedule

Dec. 30 2021, Yang Su Chinese Academy of Sciences
Time: 15:0017:00 Place: Tencent Meeting ID: 573308837 Password: 123455 Title: Geometric structure of self covering manifolds Abstract: A topological space is selfcovering if the space is a nontrivial cover of itself. I will talk about a joint work with Lizhen Qin and Botong Wang on the geometric structure of selfcovering manifolds. For example, we show that in dimension >5, a selfcovering manifold with infinite cyclic fundamental group is a fiber bundle over the circle. We also construct examples of selfcovering manifolds which are not fiber bundle over the circle, where the fundamental group contains torsion.

Dec. 16 2021, Shengkui Ye NYU Shanghai
Time: 15:0017:00 Place: Tencent Meeting ID: 533917811 Password: 123455 Title: The group of quasiisometries of the line cannot act effectively on the line Abstract: Let G be the group of orientationpreserving quasiisometries of the real line. We show that G is leftorderable, but not simple. Moreover, the group G cannot act effectively on the real line R.