Ursula Hamenstaedt
Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex cocompact in the sense of Farb and Mosher.
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We describe a procedure which verifies that a group given by generators and relators is word-hyperbolic. This procedure always works with a group which is word-hyperbolic, provided there is sufficient memory and time devoted to the problem. If the group is not word-hyperbolic, the procedure continues indefinitely. We also describe a procedure which computes the thinness of geodesic triangles in the Cayley graph of a word-hyperbolic group. Again this procedure is bound to work, given sufficient memory and time.
We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.
Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that such a manifold admits a piecewise Euclidean metric of non-positive curvature and the universal cover contains no isometrically embedded flat planes. The proof involves a mixture of computer computation and techniques from small cancellation theory.
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 longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. We define a set of polygonal words in a free group of finite rank, and prove that polygonality of the amalgamating word guarantees that the associated square complex virtually contains a $π_1$-injective closed surface. We provide many concrete examples of classes of polygonal word…
We outline a rigorous algorithm, first suggested by Casson, for determining whether a closed orientable 3-manifold M is hyperbolic, and to compute the hyperbolic structure, if one exists. The algorithm requires that a procedure has been given to solve the word problem in π_1(M).
In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered 3-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of Out$(F_n)$.
Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric length of the class is the length of the unique geodesic. By computer experiments, we investigate the distribution of the geometric length among all classes with a given word length in the pair of pants surface. Our experiments strongly suggest that the distribution is normal.
Local $C^0$-semi-rigidity of meandering-hyperbolic actions
研究蜿蜒双曲作用的局部 $C^0$-半刚性问题，通过证明其在该拓扑下的局部半刚性，将稳定性理论从利普希茨 $C^0$ 拓扑扩展到完整 $C^0$ 拓扑，进而从单一原理恢复相关作用及格的 $C^0$-局部半刚性。
Relative hyperbolicity of ascending HNN extension of groups
该研究证明了带自由因子系统的有限生成群的保系统单自同态对应的升序HNN扩张的相对双曲性，并得出有限秩自由群指数增长单自同态的升序HNN扩张相对双曲的推论，推进了几何群论相关研究。
Sublinear projection tracking in acylindrically hyperbolic groups
研究非圆柱双曲群字度量中的投影现象，对斜驶WPD元素证明相关最短投影性质，借此得到增长函数上界、构造真商群，还证明了受限子群的增长 - 余增长不等式。
Extensions of multicurve stabilizers are hierarchically hyperbolic
本文证明亏格至少为2的闭可定向曲面上多曲线稳定子的曲面群扩张为分层双曲群，回答了相关公开问题，并在附录中刻画一类分层双曲群的Morse边界。
We show that word-hyperbolic groups satisfy linear isoperimetric functions for all homotopy types of surface diagrams. This generalises the linear isoperimetric functions for disc and annular diagrams.
We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that…
We prove that all atoroidal automorphisms of $Out(F_N)$ act on the space of projectivized geodesic currents with generalized north-south dynamics. As an application, we produce new examples of non virtually cyclic, free and purely atoroidal subgroups of $Out(F_N)$ such that the corresponding free group extension is hyperbolic. Moreover, these subgroups are not necessarily convex cocompact.
We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free group $\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\mathbb{F}$. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\mathbb{F}$-trees and use this to prove that a finitely generated subgroup of $\mathrm{Out}(\mathbb{F})$ quasi-isometrically embeds into the co-surface graph if and only if it is purely atoroidal and quasi-isometrically embeds into the free factor complex. This answers a question of…
Outer automorphism groups of hyperbolic groups, bounded extensions, and hierarchical hyperbolicity
本文证明在JSJ分解的适度可定向条件下，一端双曲群的外自同构群实质上是一个层次双曲群，通过证明有限指数子群是轨道流形映射类群乘积的中心扩张且扩张具有有界欧拉类，并给出反例说明完全外自同构群不一定是HHG。
We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture in algebraic K-theory for every word-hyperbolic group G and every coefficient ring R.
Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity
本研究针对闭曲面映射类群中的格点Veech群，证明其π₁S-扩展群为分层双曲群，进而得出这类扩展群具有拟等距刚性的结论，为Veech群扩展的几何性质研究提供了新结果。
Given a finitely generated subgroup $Γ\le \mathrm{Out}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F} = F_r$, there is a corresponding free group extension $1 \to \mathbb{F} \to E_Γ \to Γ\to 1$. We give sufficient conditions for when the extension $E_Γ$ is hyperbolic. In particular, we show that if all infinite order elements of $Γ$ are atoroidal and the action of $Γ$ on the free factor complex of $\mathbb{F}$ has a quasi-isometric orbit map, then $E_Γ$ is hyperbolic. As an application, we produce examples of hyperbolic $\mathbb{F}$-extensions $E_Γ$ for whi…
Let $1\to (K,K_1)\to (G,N_G(K_1))\to(Q,Q_1)\to 1$ be a short exact sequence of pairs of finitely generated groups with $K$ strongly hyperbolic relative to proper subgroup $K_1$. Assuming that for all $g\in G$ there exists $k\in K$ such that $gK_1g^{-1}=kK_1k^{-1}$, we prove that there exists a quasi-isometric section $s\colon Q \to G$. Further we prove that if $G$ is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon-Thurston map for the inclusion $i\colonΓ_K\to Γ_G$.
The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and runs forever otherwise. In this paper, we give various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and finitely generated groups discriminat…
Relative hyperbolicity of free extensions of free groups
本文给出了自由乘自由群具有保持尖点结构的相对双曲性的充要条件，证明了当一族指数增长外自同构共享不变子群系统且补集中共轭类均指数增长时，可由该系统构造外围子群使相应扩张相对双曲。
On the type of a complex hyperbolic triangle group
本文研究由三个复对合生成的离散复双曲等距群，给出其规范形式，并改进Schwartz猜想，在参数空间中确定两个词哪个先变为椭圆型。
Extensions of Veech groups I: A hyperbolic action
本文研究闭曲面映射类群中格Veech群对应的$\pi_1S$扩展群$\Gamma$的几何，证明其为奇异欧氏乘双曲几何纤维丛的基本群，通过坍缩万有覆叠乘积区域得到$\Gamma$在双曲空间上的保几何作用，为后续证明其分层双曲性与拟等距刚性提供关键基础。
Growth gaps and exponential genericity in acylindrically hyperbolic groups
研究柱面双曲群，证明对其每个有限生成集，非WPD元素集指数增长率更小，即WPD元素指数一般，还证明了该群的增长紧性和余增长紧性。
Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity
本文通过构造在双曲平面与局部有限树乘积上几何作用的 CAT(0) HHG，给出非双自动、平坦秩 2 且单射但非 Helly 的首个反例。
In this note, we prove that a random extension of either the free group $F_N$ of rank $N\ge3$ or of the fundamental group of a closed, orientable surface $S_g$ of genus $g\ge2$ is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either Out$(F_N)$ or Mod$(S_g)$ generated by $k$ independent random walks. Our main theorem has several applications, including that a random subgroup of a weakly hyperbolic group is free and undistorted.
The algebraic structure of hyperbolic graph braid groups
本文研究了 $3$ 股图形辫群的字双曲性，证明了 sun 图的 $3$ 股辫群是自由的，并为大多数 pulsar 图的 $3$ 股辫群包含曲面子群提供了简单证明及额外的结构定理。
We consider non-elementary representations of two generator free groups in $PSL(2,\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. A word in $A$ and $B$, $W(A,B)$, is a palindrome if it reads the same forwards and backwards. A word in a free group is {\sl primitive} if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identified with the positive rational numbers. We study the geometry of palindromes and the action of $G$ in $\HH^3$ whether or not $G$ is discrete. We show that there is a {\sl core geodesic} $Ł$ in the convex hu…
We give a complete classification of complex hyperbolic $(n_1, n_2, n_3)$-triangle groups by types defined according to the ellipticity of two particular words of short length. This improves the Schwartz conjecture proved by Grossi.
This paper gives a detailed analysis of the Cannon--Thurston maps associated to a general class of hyperbolic free group extensions. Let $F_N$ denote a free groups of finite rank $N\ge 3$ and consider a \emph{convex cocompact} subgroup $Γ\le Out(F_N)$, i.e. one for which the orbit map from $Γ$ into the free factor complex of $F_N$ is a quasi-isometric embedding. The subgroup $Γ$ determines an extension $E_Γ$ of $F_N$, and the main theorem of Dowdall--Taylor \cite{DT1} states that in this situation $E_Γ$ is hyperbolic if and only if $Γ$ is purely atoroidal. Here, we give an explicit geometric d…
We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots coded by some of these families of words are hyperbolic, by showing that they are neither satellites nor torus knots and making use of Thurston's theorem. Infinite families of hyperbolic Lorenz knots are generated in this way, to our knowledge, for the first time. The techniques used…
We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including right-angled Artin groups and right-angled Coxeter groups.
Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish that when W is word hyperbolic, the limit weak order is the disjoint union of weak orders of finite Coxeter groups. We also establish, for each boundary point ξ, a natural order-preserving correspondence between infinite reduced words which "point towards" ξ, and elements of the ref…
Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.
Let $F',F$ be any two closed orientable surfaces of genus $g'>g\ge 1$, and $f:F\to F$ be any pseudo-Anosov map. Then we can "extend" $f$ to be a pseudo-Anosov map $f':F'\to F'$ so that there is a fiber preserving degree one map $M(F',f')\to M(F,f)$ between the hyperbolic surface bundles. Moreover the extension $f'$ can be chosen so that the surface bundles $M(F',f')$ and $M(F,f)$ have the same first Betti numbers.
Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary
本文用神经给出了格罗莫夫边界为谢尔宾斯基曲线或门格曲线的字双曲考克斯特群的完全刻画，主要基于对已有文献结果的组合。
Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent progress in the description of globally hyperbolic flat spacetimes showed strong link between Lorentzian geometry and Teichm{ü}ller space. We notice that Lorentzian generalisations of conical singularities are useful for the endeavours of descripting flat spacetimes, creating stronger links with hyperbolic geometry and compactifying spacetimes. In particular massive particles and extreme BTZ singular lines arise naturally. This paper is three-fold. First, prove background local properties which will be useful for futu…
We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.
Let $Σ$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $Σ$. A geodesic curve $γ$ in $Σ$ is filling, if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of $π_1(Σ)$, is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn-Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed…
Let $G$ be a word hyperbolic group. We prove that the algebraic $K$-theory groups of $\dbZ [G]$, $K_n(\dbZ[G])$, have finite rank for all $n\in \dbZ$. For a few classes of groups, we give explicit formulas for the ranks of the algebraic $K$-theory groups of their group rings.
Let $Σ_g$ be a closed hyperbolic surface of genus $g$ and let $Ham(Σ_g)$ be the group of Hamiltonian diffeomorphisms of $Σ_g$. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that $Ham(Σ_g)$ is unbounded with respect to this metric.
Tukia and Vaisala showed that every quasi-conformal map of $\R^n$ extends to a quasi-conformal self-map of $\R^{n+1}$. The restriction of the extended map to the upper half-space $\R^n \times \R^+$ is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every homogeneous negatively curved manifold decomposes as $M = N \rtimes \R^+$ where $N$ is a nilpotent group with a metric on which $\R^+$ acts by dilations. We show that under some assumptions on $N$, every quasi-symmetry of $N$ extends to a bi-Lipschitz map of $M$. The result applies to a wide class of manifolds $M$…
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