Arthur Bartels, Wolfgang Lueck
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.
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
For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the invariant trace field of the link. On the other hand, Yokota suggested a potential function of a hyperbolic knot, which gives the hyperbolicity equations and the complex volume of the knot. Applying the fact above to his theory, we explain that the potential function also gives all …
We prove that a word hyperbolic group which admits a $P_{2q+1}$-Anosov representation into $\mathsf{PGL}(4q+2, \mathbb{R})$ contains a finite-index subgroup which is either free or a surface group. As a consequence, we give an affirmative answer to Sambarino's question for Borel Anosov representations into $\mathsf{SL}(4q+2,\mathbb{R})$.
Summability for State Integrals of hyperbolic knots
该研究证明了双曲纽结$4_1$、$5_2$相关微扰级数的复苏性与Borel可求和性，提出可将Borel-Laplace重求和表示为Andersen-Kashaev状态积分组合的算法，明确了复苏结构并支持Stokes常数计算。
Borel asymptotic dimension of the Roller boundary of finite dimensional CAT(0) cube complexes
本文研究可数有限维 CAT(0) 立方体复形 Roller 紧化上的 Borel 中位数图，证明其 Borel 渐近维数不超过复形维数。
On Borel Anosov subgroups of ${\rm SL}(d,\mathbb{R})$
本文通过研究满旗流形的对极子集，在特定维数条件下确定了 ${\rm SL}(d,\mathbb{R})$ 的 Borel Anosov 子群的虚拟同构类型，并限制了可拟等距嵌入其对称空间的双曲空间。
We show that the bounded Borel class of any dense representation $ρ: G\to \PSL_n\bC$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the $3$-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $ρ: G\to \PSL_2\bC$ is uniformly separated in semi-norm from any other representation $ρ': G\to \G$ for which there is a subgroup $H\le G$ on which $ρ$ is still dense but $ρ'$ is discrete or indiscrete but stabilizes a point…
Rigidity at infinity for the Borel function of the tetrahedral reflection lattice
本文证明 Guilloux 猜想在正则理想四面体反射群情形下成立，即 Borel 函数在特征簇理想点处保持远离最大值。
Classifying group actions on hyperbolic spaces
研究能否对双曲空间上群的等距作用分类，利用博雷尔等价关系复杂性理论，给出可数群\(G\)作用的二分法结论，特殊线性群\(SL_2(F)\)满足一种情况，非初等双曲群等满足另一种情况，证明中还获有价值结果。
Let $Γ$ be a torsion-free lattice of $\text{PU}(p,1)$ with $p \geq 2$ and let $(X,μ_X)$ be an ergodic standard Borel probability $Γ$-space. We prove that any maximal Zariski dense measurable cocycle $σ: Γ\times X \longrightarrow \text{SU}(m,n)$ is cohomologous to a cocycle associated to a representation of $\text{PU}(p,1)$ into $\text{SU}(m,n)$, with $1 < m \leq n$. The proof follows the line of Zimmer' Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, it cannot exist a maximal measurable cocycle wi…
We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension $d$ for all $d\geq 4$. These examples result from applying CAT$(0)$/CAT$(-1)$ filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory.
For any triple $(W,L,ρ)$, where W is a closed connected and oriented 3-manifold, L is a link in W and $ρ$ is a flat principal B-bundle over W (B is the Borel subgroup of $SL(2,\mc)$), one constructs a $\Dd$-scissors congruence class $\cG_{\Dd}(W,L,ρ)$ which belongs to a (pre)-Bloch group $\Pp (\Dd)$. The class $\cG_{\Dd}(W,L,ρ)$ may be represented by $\Dd$-triangulations $\Tt=(T,H,\Dd)$ of $(W,L,ρ)$. For any $\Tt$ and any odd integer $N>1$, one defines a ``quantization'' $\Tt_N$ of $\Tt$ based on the representation theory of the quantum Borel subalgebra $\Ww_N$ of $U_q(sl(2,\mc))$ specialized …
Manifold classification from the descriptive viewpoint
本文从描述集合论角度研究流形和李群离散子群的分类问题，揭示了流形同胚问题的Borel复杂性及双曲流形等距关系的分类特性。
Measurable boundary maps and Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary
本文研究非博雷尔阿诺斯群在弗斯滕伯格边界上的帕特森-苏利文测度，探讨其存在性、唯一性和ergodic性，以及临界指数的严格凸性，并建立阿诺斯群的熵刚性结果。
We generalize the proof of the Farrell-Jones conjecture for CAT(0)-groups to a larger class of groups in particular also containing all hyperbolic groups. This way we give a unified proof for both classes of groups.
CAT(0) property and hyperbolicity of affine cactus groups
本文证明了所有度数的仿射仙人掌群都是CAT(0)群，并进一步证明了三次仿射仙人掌群AJ_3是双曲群。
We prove an explicit equivalence between various hyperbolic type properties for quasi-geodesics in CAT(0) spaces. Specifically, we prove that for X a CAT(0) space and $γ$ a quasi-geodesic, the following four statements are equivalent and moreover the quantifiers in the equivalences are explicit: (i) $γ$ is S-Slim, (ii) $γ$ is M-Morse, (iii) $γ$ is (b,c)-contracting, and (iv) $γ$ is C-strongly contracting. In particular, this explicit equivalence proves that for $f$ a (K,L)-quasi-isometry between CAT(0) spaces, and $γ$ a C-strongly contracting (K',L')-quasi-geodesic, then $f(γ)$ is a C'(C,K,L,K…
We prove that every finite-volume hyperbolic 3-manifold M with p > 0 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K. Moreover, (a) the universal cover of M endowed with the CAT(0) metric is a union of Euclidean half-spaces, glued together by identifying Euclidean polygons in their bounding planes by pairwise isometry (b)each cusp of M in the CAT(0) metric is a non-singular metric product of a (Euclidean) cusp torus and a half-line (c) all metric singularities are concentrated on the 1-skeleton of K, with cone angles a mult…
We describe a simple locally CAT(0) classifying space for extra extra large type Artin groups (with all labels at least 5). Furthermore, when the Artin group is not dihedral, we describe a rank 1 periodic geodesic, thus proving that extra large type Artin groups are acylindrically hyperbolic. Together with Property RD proved by Ciabonu, Holt and Rees, the CAT(0) property implies the Baum-Connes conjecture for all XXL type Artin groups.
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of "n-tracks of length n", which is a structure like Z+Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts effectively, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is "weakly special", then the above question is answered affirmatively.
Mahan Mitra (Mj) proved Cannon--Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon--Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston maps do not exist for infinite infinite-index normal CAT(0) subgroups with isolated flats in non-hyperbolic CAT(0) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have …
The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and we show that contracting isometries are strongly related to isolated points at infinity, when the complex is locally finite. This boundary turns out to appear naturally in the context of Guba and Sapir's diagram groups, and we apply our main criterion to determine precisely when an element of a diagram group induces a contracting isometry on the associated …
We prove that in CAT(0) spaces a quasi-geodesic is Morse if and only if it is contracting. Specifically, in our main theorem we prove that for $γ$ a quasi-geodesic in a CAT(0) space X, the following four statements are equivalent: (i) $γ$ is Morse, (ii) $γ$ is (b,c)--contracting, (iii), $γ$ is strongly contracting, and (iv) in every asymptotic cone $X_ω,$ any two distinct points in the ultralimit $γ_ω$ are separated by a cutpoint. As a corollary, we provide a converse to the usual Morse stability lemma in the CAT(0) setting. In addition, as a warm up we include an alternative proof of the fact…
We prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of …
The main result of this paper is that given a group $G$ acting geometrically by isometries on a CAT(0) space $X$ and a cyclic subgroup $H$ of $G$ generated by a rank-1 isometry of $X$, $H$ has bounded packing in $G$. We give two proofs of this result. The first one is by a characterization of rank-$1$ isometries by Hamenstadt. The second proof follows directly from some results of Dahmani-Guirardel-Osin and Sisto. Then using Mihailova's construction, we show the existence of a finitely generated subgroup of the direct product of two free groups $\mathbb F_2\times \mathbb F_2$ without the bound…
Suppose a group $G$ is relatively hyperbolic with respect to a collection $\PP$ of its subgroups and also acts properly, cocompactly on a $\CAT(0)$ (or $δ$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary $\partial(G,\PP)$. The $\CAT(0)$ structure provides a different boundary at infinity $\partial X$. In this article, we examine the connection between these two spaces at infinity. In particular, we show that $\partial (G,\PP)$ is $G$--equivariantly homeomorphic to the space obtained from $\partial X$ by identifying the peripheral limit points of the sam…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。