arXiv 0907.37552009-07math.GTmath.DG

Smooth (non)rigidity of cusp-decomposable manifolds

T. Tam Nguyen Phan

备注：: 13 pages, 1 figure. Accepted for publication in Comment. Math. Helv. arXiv admin note: substantial overlap with arXiv:1105.5205

英文摘要：

We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, locally symmetric, negatively curved manifolds with cusps. We prove that the group of outer automorphisms of the fundamental group of such a manifold is an extension of an abelian group by a finite group. Elements of the abelian group are induced by diffeomorphisms that are analogous to Dehn twists in surface topology. We also prove that the outer automophism group can be realized by a group of diffeomorphisms of the manifold.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Quasi-Isometric Rigidity of Cusp-Decomposable Manifolds

In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguyên Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometric rigidity for fundamental groups of cusp-decomposable manifolds.

2017-06-09math.GT

On the quasi-isometric rigidity of chambers and walls in cusp-decomposable manifolds

A cusp-decomposable manifold is a manifold constructed from a finite number of complete, negatively curved, finite volume manifolds and identifying the boundaries of truncated cusps by diffeomorphisms. Using properties of the electric space of the universal cover of cusp-decomposable manifolds, we show that the inclusion of walls and pieces induces quasi-isometric embeddings. We also show that isomorphisms between fundamental groups of higher graph manifolds preserve the decomposition into pieces.

2020-10-08math.GT

Smooth (non)rigidity of piecewise rank one locally symmetric manifolds

We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with $π_1$-injective cusps. We prove smooth (self) rigidity for this class of manifolds in the case where the gluing preserves the cusps' homogeneous structure. We compute the group of self homotopy equivalences of such a manifold and show that it can contain a normal free abelian subgroup and thus, can be infinite. Elements o…

2011-05-30math.DGmath.GT

The infinitesimal projective rigidity under Dehn filling

To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally projectively rigid relative to the boundary, then infinitely many Dehn fillings are projectively rigid. We analyze in more detail the figure eight knot and the Withehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings.

2011-09-28math.DGmath.GT

Constructing convex projective 3-manifolds with generalized cusps

We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling which types of cusp occur. Using these methods we produce the first known example of a 1-cusped hyperbolic 3-manifold that admits a convex projective structure with a type 2 cusp. We also use these techniques to produce new 1-cusped manifolds that admit a convex projective structu…

2020-10-26math.DGmath.GT

有限扩张刚性通过带凹口的双曲3流形的Dehn填充 witnessed

Profinite rigidity witnessed by Dehn fillings of cusped hyperbolic 3-manifolds

本文通过研究带凹口的双曲3流形的Dehn填充，证明了某些流形在所有紧致定向3流形中是有限扩张刚性的，并改进了双曲结或双曲型卫星结的有限扩张刚性判据。

2026-03-16math.GRmath.GT

更多相关论文收起更多论文

尖点双曲三维流形的 systole、内半径与刚性

Systole, inradius and rigidity of cusped hyperbolic 3-manifolds

Univ Paris Est Creteil（巴黎东克雷泰伊大学）；CNRS（法国国家科学研究中心）；LAMA（拉马实验室）；Univ Gustave Eiffel（古斯塔夫·埃菲尔大学）

建立了有限体积双曲三维流形中 systole 和内半径与体积之间的最优不等式，并确定了极值流形，包括八之字结补及其姐妹流形、Weeks-Matveev-Fomenko 流形等。

2026-06-04math.GT

关联复形的实现与 proﬁnite 刚性

Cubical residual complexes and profinite rigidity of real hyperbolic lattices

本文通过可除填充与完备胞腔关联复形实现带标记 proﬁnite 群，证明离散与完备作用的外自同构群一致，并应用于 Kleinian 群获得 PL 实现与边界子群对应。

2026-10-02math.DGmath.GT

Rigidity Properties for Hyperbolic Generalizations

We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditio…

2019-07-15math.GRmath.GT

Profinite 刚性与几何收敛

Profinite rigidity and geometric convergence

本文证明 profinitely rigid 有限体积双曲流形在几何拓扑下闭合并借助 bubble-drilling 构造推广到一大类带尖点双曲流形，涵盖 Whitehead link 与 Borromean ring 补空间。

2025-01-04math.GRmath.GT

On finite marked length spectral rigidity of hyperbolic cone surfaces and the Thurston metric

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichmüller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. We compare such a Teichmüller space with the Teichmüller space of complete hyperbolic surfaces with punctures, by sho…

2017-03-06math.CVmath.GTmath.MG

Rigidity of high dimensional graph manifolds

We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic manifold with toric cusps. The various pieces are attached together via affine maps of the boundary tori. We require all the hyperbolic factors in the pieces to have dimension at least 3. Our main goal is to study this class of graph manifolds from the viewpoint of rigidity theory. We show that, in high dimensions, the Borel conjecture holds for our graph manifol…

2013-12-21math.DGmath.GRmath.GT

Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds

Let W be a compact manifold and let ρbe a representation of its fundamental group into PSL(2,C). The volume of ρis defined by taking any ρ-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a non-compact (cusped) manifold M, but he did not prove the volume is well-defined in all cases. We prove here that the volume of a representation is always well-defined and depe…

2003-05-19math.GT

具有收缩元素的群中的标记长度谱刚性

Marked length spectrum rigidity in groups with contracting elements

本文研究了粗几何背景下的标记长度谱刚性问题，证明具有收缩性质的群作用在相同标记长度谱下轨道映射为粗等距，并推广了双曲群的相关结论。

2025-05-03math.GRmath.GT

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficien…

2008-11-15math.GRmath.GT

关于双曲4流形的Margulis尖点

On Margulis cusps of hyperbolic 4-manifolds

本文研究4维双曲流形中由无理螺旋平移生成的Margulis区域的边界函数，给出其渐近上界、下界及最优界，并证明对应Margulis尖点的双Lipschitz刚性。

2013-07-30math.DSmath.GT

Entropy and finiteness of groups with acylindrical splittings

We prove that there exists a positive, explicit function $F(k, E)$ such that, for any group $G$ admitting a $k$-acylindrical splitting and any generating set $S$ of $G$ with $\mathrm{Ent}(G,S)<E$, we have $|S| \leq F(k, E)$. We deduce corresponding finiteness results for classes of groups possessing acylindrical splittings and acting geometrically with bounded entropy: for instance, $D$-quasiconvex $k$-malnormal amalgamated products acting on $δ$-hyperbolic spaces or on $CAT(0)$-spaces with entropy bounded by $E$. A number of finiteness results for interesting families of Riemannian or metric …

2018-04-12math.DGmath.GTmath.MG

Geodesic Planes in Geometrically Finite Manifolds

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…

2021-10-09math.DSmath.GT