Topics on hyperbolic geometry 讨论班15:00-17:00

[ 研究院访问学者将在6月20日至7月20日期间组织一个不定期的讨论班。讨论班主要围绕hyperbolic geometry (and low-dimensional topology)开展约八场90分钟的学术报告。本次研讨会旨在学者间加强沟通了解该领域的研究进展。]



主讲人:刘毅 (北京大学)
题目:Profinite completion and surface mapping classes
摘要:In two talks, I plan to discuss recent studies on 2- or 3-manifold groups and their profinite completions. The profinite completion of a group refers to the inverse limit of all the finite quotient groups of that group. The topic of profinite completion has attracted increasing interest in low-dimensional topology during the past decade. I will try to make my two talks focus on relatively different directions. In the first talk, I will discuss surface mapping classes and their induced action on the profinite surface group. The next talk will be about hyperbolic 3-manifolds and the profinite completion of their fundamental groups.
主讲人:马继明 (复旦大学)
题目:Schwartz's complex hyperbolic surface

Let $G(4, 7)$ be a certain finitely presented group, in a celebrated paper of Schwartz in 2003,

R. Schwartz considered a representation of $G(4, 7)$ into the isometry group of the complex hyperbolic plane. R. Schwartz determined the 3-manifold at infinity of the quotient complex hyperbolic surface via a sophisticated method. More precisely, the 3-manifold at infinity is a closed hyperbolic 3-orbifold with underlying space the 3-sphere and whose singularity locus is a two-components link equipped with a $\mathbb{Z}_2$-cone structure. In this talk, we will show the representation above is faithful, and determine the 4-dimensional topology of the complex hyperbolic surface via the handle structure.

主讲人:刘毅 (北京大学)
题目:Profinite completion and hyperbolic 3-manifolds
摘要:In this talk, I will discuss recent advances in combining profinite group theory with hyperbolic 3-manifold topology. There are topological properties which are also profinite properties, such as fiberedness, and the Thurston norm on the first cohomology. We will give an introduction to results in that direction.
主讲人;叶圣奎 (上海纽约大学)
题目:Length functions on groups and applications to actions on Gromov hyperbolic spaces

A length function l on a group G is a real-valued function that is conjugation-invariant, homogenous, and subadditive with respect to commuting elements. Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on CAT(0) spaces and Gromov delta-hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, smooth entropy, dynamical degrees of birational maps and so on. In this talk, we will briefly review the properties of length functions and discuss applications to group actions on Gromov hyperbolic spaces. In particular, we will show that any (rough) isometric action of a finite-index subgroup of SL(n,R),n>2, (R is a ring of algebraic integers in a number field) on a Gromov hyperbolic space must have a fixed point in a X or its Gromov boundary. 

主讲人:马继明 (复旦大学)
题目:Compact hyperbolic Coxeter 4-polytopes with 8-facets

Unlike the spherical and parabolic cases, complete classification regarding hyperbolic Coxeter polytopes of finite volume is far from being obtained. Poincare and Andreev addressed this problems in dimensions 2 and 3, respectively.  In dimensions larger than or equal to four, complete classifications of Coxeter polytopes are achieved scatteredly only in the cases of simplexes, n-polytopes of finite volume with  n+2 facets and bounded n-polytope with n+3 facets, etc. We obtain the complete classification for compact hyperbolic Coxeter 4-polytopes with 8 facets. This is  joint work with Fangting Zheng.

主讲人:叶圣奎 (上海纽约大学)
题目:Length functions on groups and applications to actions on Gromov hyperbolic spaces

In the second talk, we will continue to discuss the (rough) action of a finite-index subgroup of SL(n,R),n>2, (R is a ring of algebraic integers in a number field) on a Gromov hyperbolic space.  After proving the general fixed-point property, we will discuss the actions on manifolds of pinched negative curvatures, for which more rigid results can be proved.  

题目:The most symmetric Heegaard splittings
摘要:It is known that in the orientable category the maximum order of finite group actions on genus g>1 Heegaard splittings is 12(g-1). We will give a classification of the actions realizing this bound in the meaning of Thurston’s geometrization of 3-orbifolds. As a useful tool, alternating trivalent graphs and its relation with geometric 3-orbifolds will also be introduced.


T是一个双曲单孔环面(简称为双曲环面)。这里的“孔”可以是开口(设边界测地线的长度为λ>0)或尖点。任取该双曲结构的和乐表示的提升,使之成为曲面的基本群到SL(2,R)的表示,可知围绕孔的简单闭曲线所对应的矩阵的迹为 τ = -2cosh(λ/2), 或 -2.

记 μ=τ+2, 则有 μ≤0. 可以选择和乐表示的提升,使得T上每条非边界的简单闭测地线(长度为L>0)的迹都是正的,从而等于2cosh(L/2) > 2. 选定T上三条简单闭测地线,它们两两相交一次;设它们的迹分别为 x>2, y>2, z>2; 则 x, y, z 满足所谓几何Markoff方程

x^2 +y^2 +z^2 –xyz = μ.

T上每条闭测地线的迹是x, y, z的多项式。


Teich_{1,1}(μ) = { (x, y, z) | x>2, y>2, z>2, x^2 +y^2 +z^2 –xyz = μ }.

则最对称的双曲环面T对应于参数组 (3+x, 3+x, 3+x), 其中x0. 设在T的和乐表示的适当提升之下,选定的三条两两相交一次的简单闭测地线的SL(2,R)矩阵分别为A, B, A^{-1}B, 它们的迹都等于3+x. T上一条允许重复多圈的简单闭测地线所对应的正字W_{m, n} nAmB组成(不妨要求m≤n),迹为M(m, n).

对于多项式f, g, 我们记 f >>> g, 如果差 f - g 不是零多项式,且系数非负。


(A, B写出的任何正字W的迹,作为x的多项式,其系数都是正的。我们进一步猜测该多项式的系数序列具有对数凹性。

( W 是用nAmB写出的字(其中m≤n),而且W既不是简单闭曲线的字,也不是闭曲线A^n B^m的字。则有如下迹多项式不等式:

tr(W_{m, n}) <<< tr(W) <<< tr(A^n B^m).

 (迹多项式 M(m, n) 具有如下单调性(其中m≤n)

 (2+x) M(m-1, n) <<< M(m, n);

② (2+x) M(m, n) <<< M(m, n+1);

③ M(m-1, n+) >>> M(m, n).

x=0的情形,这给出M. Aigner在书Markov’s Theorem and 100 Years of the Uniqueness Conjecture (Springer, 2013)中所猜测的Markoff数的单调性。


 M(m-2, n+2) + M(m, n) >>> 2 M(m-1, n+1). 
题目:On closed geodesics on a pair of pants

We first discuss known results of C.M. Baribaud (1999) on closed geodesics on a pair of pants, and then propose new research problems.