Boundaries of Dehn fillings

Recognizing a relatively hyperbolic group by its Dehn fillings

Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic $3$-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of non-elementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.

Generalized torsion and Dehn filling

A generalized torsion element is a non-trivial element such that some non-empty finite product of its conjugates is the identity. We construct a generalized torsion element of the fundamental group of a 3-manifold obtained by Dehn surgery along a knot in the 3-sphere.

Farrell-Jones via Dehn fillings

Following the approach of Dahmani, Guirardel and Osin, we extend the group theoretical Dehn filling theorem to show that the pre-images of infinite order elements have a certain structure of a free product. We then apply this result to show that groups hyperbolic relative to residually finite groups satisfying the Farrell-Jones conjecture, they satisfy the Farrell-Jones conjecture.

Quasiconvexity and Dehn filling

We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with a recent result of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hyperbolic 3-manifolds, a result first proved by Wise. Additionally, we explain how the results of [2, Appendix A] can be generalized to the relative setting to control the relative height of relatively quasiconvex subgroups under appropriate Dehn fillings.

Homological and homotopical Dehn functions are different

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions, but prior to this work there were no known examples of finitely-presented groups for which the two functions differ. This paper gives the first such examples, constructed …

使用代数Dehn填充区分Gromov-Thurston流形

Distinguishing Gromov-Thurston manifolds using algebraic Dehn fillings

通过将基本群描述为相对双曲群的虚拟Dehn填充，发展出区分Gromov-Thurston流形同伦型的判据。

基于群论的Dehn填充视角：Property P猜想及其扩展

Group theoretic perspective on Dehn fillings: Property P conjecture and beyond

本文从群论角度提出研究Dehn填充的新视角，探讨Property P猜想及其扩展。

半单李群中的 Dehn 填充

Dehn filling in semisimple Lie groups

将Thurston双曲Dehn填充定理推广至任意秩半单李群，证明扩展几何有限子群的形变保持性质，并给出相对Anosov子群形变保持Anosov的准则，适用于凸射影几何高秩例子。

区分群中共轭相关的填充不变量

Distinguishing Filling Invariants Associated to Conjugacy in Groups

本研究针对群中共轭相关的填充不变量，补充了环形Dehn函数理论的多样群例，证明其与Dehn函数、共轭子长度函数三者相互独立，任意两者都无法决定第三者。

Convex projective generalized Dehn filling

For $d=4, 5, 6$, we exhibit the first examples of complete finite volume hyperbolic $d$-manifolds $M$ with cusps such that infinitely many $d$-orbifolds $M_{m}$ obtained from $M$ by generalized Dehn filling admit properly convex real projective structures. The orbifold fundamental groups of $M_m$ are Gromov-hyperbolic relative to a collection of subgroups virtually isomorphic to $\mathbb{Z}^{d-2}$, hence the images of the developing maps of the projective structures on $M_m$ are new examples of divisible properly convex domains of the projective $d$-space which are not strictly convex, in cont…

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。

具有层次超几何群的漏斗空间，及其在德恩填充商中的应用

Cusped spaces for hierarchically hyperbolic groups, and applications to Dehn filling quotients

本文提出一种构造方法，同时产生相对超几何群的漏斗空间和准等距于泰克米勒度量的空间。通过此方法研究了各种群的德恩填充类似商，包括穿孔球的映射类群。特别地，证明了五孔球的映射类群（或四strand辫群）有无限超几何商，且不与其他球面映射类群（或辫群）的超几何商同构。

Dehn filling Dehn twists

Let $Σ_{g,p}$ be the genus--$g$ oriented surface with $p$ punctures, with either $g>0$ or $p>3$. We show that $MCG(Σ_{g,p})/DT$ is acylindrically hyperbolic where $DT$ is the normal subgroup of the mapping class group $MCG(Σ_{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $Σ_{g,p}$ for suitable $K$. Moreover, we show that in low complexity $MCG(Σ_{g,p})/DT$ is in fact hyperbolic. In particular, for $3g-3+p\leq 2$, we show that the mapping class group $MCG(Σ_{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on s…

Dehn填充的$L^2$-Betti数

$L^2$-Betti numbers of Dehn fillings

本文研究群论Dehn填充的$L^2$-Betti数，证明了对于一大类virtually special群，其充分深的Dehn填充保持$L^2$-Betti数不变，并应用于验证Singer猜想、建立虚拟纤维化准则及构造具有奇异子群的双曲群新例子。

Deciding Isomorphy using Dehn fillings, the splitting case

We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if…

Nontrivial elements in a knot group which are trivialized by Dehn fillings

Let K be a nontrivial knot in the 3-sphere with the exterior E(K), and u in G(K), the fundamental group of E(K), a slope element represented by an essential simple closed curve on the boundary of E(K). Since the normal closure of u in G(K) coincides with that of the inverse of u, and u and its inverse u correspond to a slope r, a rational number or 1/0, we write << r >> = << u >>. The normal closure << u >> describes elements which are trivialized by r-Dehn filling of E(K). In this article, we prove that << r_1 >> =<< r_2 >> if and only if r_1 = r_2, and for a given finite family of slopes S =…

Peripheral fillings of relatively hyperbolic groups

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $π_1(M)$. The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and the group $G$ is approximated (in an algebraic sense) by its quotients obtain…

Dehn filling和knot group II: persistent elements的普遍性

Dehn filling and the knot group II: Ubiquity of persistent elements

研究非平凡结的Dehn填充下持久元素的存在性，证明每非平凡结群中存在无限多个持久元素，且这些元素在群结构中广泛分布。

Dehn filling in relatively hyperbolic groups

We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, "preferred paths", is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2πTheorem in the …

Word hyperbolic Dehn surgery

The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We establish an extension of the Thurston-Gromov $2π$ theorem by showing that if each filling slope has length more than six, then the resulting 3-manifold has all the above properties. We also give a combinatorial version of the $2π$ theorem which relates to angled ideal triangulations. We apply these techniques by studying surgery along alternating links.

A coarse embedding theorem for homological filling functions

We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embedding into a hyperbolic group of geometric dimension $2$, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension $2$, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.

Dehn填充与纽结群I：实现性质

Dehn filling and the knot group I: Realization Property

本文研究了双曲纽结群中Dehn填充对元素平凡化的实现性质，证明在无例外手术等一般条件下任意有限斜率族均可被某元素实现，并指出该性质非无条件成立。

实理想点、Conway球面与左可序Dehn填充

Real ideal points, Conway spheres, and left-orderable Dehn fillings

本文研究含本质Conway球面的纽结外部的$SL_2(\boldsymbol{C})$特征簇的实理想点，结合相关结果验证了部分纽结大斜率Dehn填充的$L$空间猜想，还对三类纽结族及28个普查外部完成了相关计算验证。

Filling loops at infinity in the mapping class group

We study the Dehn function at infinity in the mapping class group, finding a polynomial upper bound of degree four. This is the same upper bound that holds for arbitrary right-angled Artin groups.

Fillings, finite generation and direct limits of relatively hyperbolic groups

We examine the relationship between finitely and infinitely generated relatively hyperbolic groups, in two different contexts. First, we elaborate on a remark from math.GR/0601311, which states that the version of Dehn filling in relatively hyperbolic groups proved in math.GR/0510195, allowing infinitely generated parabolic subgroups, follows from the version with finitely generated parabolics. Second, we observe that direct limits of relatively hyperbolic groups are in fact direct limits of finitely generated relatively hyperbolic groups. We use this (and known results) to derive some consequ…

相对填充函数的拟等距不变性

Quasi-isometry invariance of relative filling functions

本研究证明有限生成群与子群配对的相对Dehn函数具有拟等距不变性，揭示拟等距对几乎反正规性、锥化Cayley图精细性的保持作用，明确组合Dehn函数良定义的等价条件，给出双曲嵌入子群相关结论及Baumslag-Solitar群的特例结果。

Residual finiteness, QCERF, and fillings of hyperbolic groups

We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling.

Dehn 填充、等变同调与 Baum-Connes 猜想

Dehn fillings, equivariant homology, and the Baum-Connes conjecture

本文建立 Cohen-Lyndon 三元组与等变同调的联系，证明有限圈积系数的 Baum-Connes 猜想对离散双曲群及一大类相对双曲群（含实秩一简单 Lie 群中的格）成立。

Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for …

三维映射环面的整填充体积、复杂度与整单纯形体积

Integral filling volume, complexity and integral simplicial volume of 3-dimensional mapping tori

本文证明 Dehn 扭转的整填充体积为零，刻画整填充体积消失的映射类，并据此说明三维映射环面的复杂度与整单纯形体积不具 Lipschitz 等价性。