Any 3-manifold 1-dominates at most finitely many geometric 3-manifolds
Shicheng Wang, Qing Zhou
本文证明了任意闭可定向3-流形1-支配至多1000个几何3-流形。
In this paper, we proved that any closed orientable 3-manifold 1-dominates at most finitely many geometric 3-manifolds.
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This is the second of a Series of three papers, the first one published in Geom Dedicata 167 p. 91-121 (2013), proving that all finitely presented groups are QSF.
A closed 4-manifold (or, more generally, a finite $PD_4$-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a $PD_3$-complex.
Non-zero degree maps between closed orientable three-manifolds
本文讨论了如下问题：给定一个闭可定向三维流形M，是否至多存在1000个由M 1-支配的闭可定向三维流形？我们解决了这个问题的类封闭定向图流形。更确切地说，本文的主要结果是：任何闭可定向图流形1-支配至多1000个满足Poincare-Thurston几何化猜想的可定向闭三元流形。为了证明这个结果，我们给出了一个更一般的Haken流形定理，即任何闭可定向三流形M1-支配至多1000个Haken流形，其Gromov单纯体积足够接近M的Gromov单纯体积。
This paper gives a quick overview of the author's recent result that all finitely presented groups are QSF.
This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of easy groups. As we will explain, these properties are central in the theory of discrete groups seen from a topological viewpoint at infinity. Also, we shall outline the main steps of the achievements obtained in the last years around the very general question whether or not all finitely presented groups satisfy these conditions.
Hyperbolic spaces with geometric and geometrically finite quasi-actions are symmetric
研究证明，若一个度量空间与有限生成群及具有上空的有限生成群空间准等距，则该空间与一维对称空间或实直线准等距。
We present an algorithm which given a presentation of a group $G$ without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant $κ$, outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decides if $G$ admits an essential free decomposition, the second is an algorithm which; if $G$ is relatively hyperbolic; decides if it admits an essential elementary splitting.
We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a given compact 3-manifold belong, up to homeomorphism, to a finite collection of compact 3-manifolds. We show also that any closed orientable 3-manifold dominates only finitely many integral homology spheres and any compact 3-manifolds orientable 3-manifold dominates only finitel…
Any action of a group $Γ$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $Γ$. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic $3$-manifolds. That is, if two homotopy equivalent hyperbolic manifolds with infinite volume and without parabolic cusps have different geometrically infinite end invariants, then they define a $2$ dimensional subspace of bounded cohomology. Our techniques apply to…
In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that degree onto only finitely many homeomorphically distinct orientable closed 3-manifolds.
A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely covered by a product of two aspherical surfaces does not support (transitive) Anosov diffeomorphisms.
Geometric Diffeomorphism Finiteness in Low Dimensions and Homotopy Group Finiteness
我们的主要结果是：对于任意给定的数C和D，m<7维单连通闭光滑流形类只包含2/3个单同态类型，其中m<7维单连通闭光滑流形类的截面曲率以C为绝对值有界，直径以D为上一致有界.因此，在这些维度中，Cheeger定理中的体积的正下界可以被一个纯粹的拓扑条件--单连通性所取代。在4维中，这里只需要非零的欧拉示性数而不是单连通性。作为一个拓扑推论，我们得到：当k+l<7时，在给定的光滑闭l-流形上，只有1/2个主$T^k$丛具有单连通的非同构全空间.此外，对于任意给定的数C和D以及任意维数m，证明了对于任意自然数i，对于一个单连通闭m-流形的第i个同伦群，对于每个自然数i，总是有至多至多1/2个同构的可能性，该闭m-流形允许一个度量，其截面曲率以C为绝对值，直径以D为上界.
Geometric limits of knot complements
证明了嵌入到$\BS^3$中的具有单拓扑端点的双曲生成基本群的完备3-流形是$\BS^3$中双曲纽结补序列的几何极限.特别地，我们得到了双曲纽结补的存在性，其中包含球的任意大的半径。我们还证明了具有两个凸余紧端点的完备双曲3-流形不可能是$\BS^3$中纽结补的几何极限。
An introduction to semistability in geometric group theory
本文介绍几何群论中无穷远处半稳定性的概念与两类证明技术，并以组合定理为例证明了相对多循环子群的双曲群的无穷远处半稳定性。
The discreteness problem for finitely generated subgroups of $PSL(2,\mathbb{R})$ and $PSL(2,\mathbb{C})$ is a long-standing open problem. In this paper we consider whether or not this problem is decidable by an algorithm. Our main result is that the answer depends upon what model of computation is chosen. Since our discussion involves the disparate topics of computability theory and group theory, we include substantial background material.
Quasiconformal Rigidity of Negatively Curved Three Manifolds
本文研究了具有截面曲率$-b^2\le K\le -1$的无限体积三维流形的刚性，并讨论了它的生成基本群。特别地，我们将具有空耗散集的非线性生成基本群的Sullivan拟共形刚性推广到负变曲率三维流形。我们还推广了Hamenstädt或最近Besson-Courtois-Gallot的刚性，3-流形具有无限体积和几何无限的基本群。
Let $π$ be a finitely presented group. If h is a non trivial homology class in Hn($π$; Z), a theorem of Gromov (see [Gro83], 6) asserts the existence of regular geometric cycles which represent h, whose relative systolic volume is as close as desired to the systolic volume of h, in which we can control the volume of balls of radius less than half of the cycle's relative systol. The aim of this note is to explain and provide a complete proof of this result.
We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank $1$ cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property was introduced in the recent work of McMullen, Mohammadi and Oh. Nonetheless, in the setting of geometrically finite groups whose limit sets are circle packings, we derive 2 density criteria for non-closed geodesic plane immersions, and show that closed immersions give rise to surfaces with finitely generated fundamental groups. We also obtai…
Dehn functions of subgroups of products of free groups. Part I: Uniform upper bounds
本文证明三个自由群直积的有限展示子群及 n 个自由群直积中 $\mathcal{F}_{n-1}$ 型子群的 Dehn 函数均以 $N^9$ 为上界，正面回答 Dison 问题并为 Bridson 猜想提供新证据。
In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric structures on orbifolds, and Hilbert geometry. The main examples of finitely generated groups for us will be Fuchsian groups, 3-manifold groups and Coxeter groups.
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