Jan Dymara, Damian Osajda
We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. Corollary: the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.
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By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.
By Andreev theorem acute-angled polyhedra of finite volume in a hyperbolic space $\mathbb H^{3}$ are uniquely determined by combinatorics of their 1-skeletons and dihedral angles. For a class of compact right-angled polyhedra and a class of ideal right-angled polyhedra estimates of volumes in terms of the number of vertices were obtained by Atkinson in 2009. In the present paper upper estimates for both classes are improved.
The minimal covolume right-angled Coxeter group in hyperbolic 3-space
本文证明由特定直角三角双锥体面反射生成的直角Coxeter群在双曲3空间中具有最小余体积，该群为算术群，余体积等于卡塔兰常数。
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Löbell polyhedra, a class of polyhedra which generalizes the dodecahedron. Furthermore, these combinatorial operations are shown to have geometric realizations which are volume decreasing. This allows for an organization of the volumes …
For arbitrary integer n, we describe a large class of right-angled Coxeter systems for which the visual baundary (of the corresponding Coxeter-Davis complex) is homeomorphic to the n-dimensional Sierpiński compactum. We also provide a necessary and sufficient condition for a planar simplicial complex L under which the right-angled Coxeter system whose nerve is L has the visual boundary homeomorphic to the Sierpiński curve.
In this note, we improve Nikulin's inequality in the case of right-angled hyperbolic polyhedra. The new inequality allows to give much shorter proofs of the known dimension bounds. We also improve Nonaka's lower bound on the number of ideal vertices for right-angled hyperbolic polyhedra.
To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume of the alternating link. We give an explicit procedure to compute the right-angled volume from any alternating link diagram, and prove that it is a new lower bound for the hyperbolic volume of the link.
We consider the control problem where, given an orthonormal tangent frame in the hyperbolic plane or three dimensional hyperbolic space, one is allowed to transport the frame a fixed distance $r > 0$ along the geodesic in direction of the first vector, or rotate it in place a right angle. We characterize the values of $r > 0$ for which the set of orthonormal frames accessible using these transformations is discrete. In the hyperbolic plane this is equivalent to solving the discreteness problem for a particular one parameter family of two-generator subgroups of $PSL_2(\mathbb{R})$. In the three…
For a hyperbolic link K in the thickened torus with no bigons, we show that there is a decomposition of the complement of a link L, obtained from augmenting K, into torihedra. We further decompose the torihedra into angled pyramids and finally angled tetrahedra. These fit into an angled structure on a triangulation of the link complement, and thus by [5], this shows that L is hyperbolic.
As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space $\mathbb{H}^n$ has at least one cusp for $n\geq 5$. We obtain non-trivial lower bounds on the number of cusps of such polyhedra. For example, right-angled polyhedra of finite volume must have at least three cusps for $n=6$. Our theorem also says that the higher the dimension of a right-angled polyhedron becomes, the more cusps it must have.
In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of the boundaries are arbitrarily connected. We also prove a combination theorem for certain amalgams of CAT(0) groups to act on spaces with non-path connected visual boundaries. We apply this theorem to some right-angled Artin groups.
An algorithm for determining the list of smallest volume right-angled hyperbolic polyhedra in dimension 3 is described. This algorithm has been implemented on computer using the program Orb to compute volumes, and the first 825 polyhedra in the list have been determined.
We show that the Morse boundary of a right-angled Coxeter group may contain embedded circles that do not arise as the boundary of a Morse Fuchsian subgroup visible in the defining graph.
This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique. The method involves subdividing M into ideal tetrahedra and solving a system of gluing equations to find hyperbolic shapes for the tetrahedra. The gluing equations decompose into a linear and non-linear part. The solutions to the linear equations form a convex polytope A. The solution to the non-linear part (unique if it exists) is a critical point of a certain volume functional…
Surface Quotients of Right-Angled Hyperbolic Buildings
本文研究 Fuchsian 建筑的曲面商，证明一大类建筑存在自同构群离散子群使其商为无边界紧致曲面，并给出基于建筑对称性的必要条件。
This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements.
Automorphism and outer automorphism groups of Right-angled Artin groups are not relatively hyperbolic
本文证明定义图至少含3个顶点的右角Artin群的自同构群，以及非实质上同构于单顶点或不连通定义图的外自同构群，均不是相对双曲的。
In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D. Mostow. In the case of buildings of dimension 2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewner p…
In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with only finite (or usual) vertices, and for finite volume polyhedra with vertices of both types.
This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph to its defining graph. More generally, we present a method to construct amalgamated free products of CAT(0) groups with totally disconnected Morse boundaries that act geometrically on CAT(0) spaces that have a treelike block decomposition.
Local Connectivity of Right-angled Coxeter group boundaries
本文研究直角 Coxeter 群所作用 CAT(0) 空间边界的局部连通性，给出由定义图刻画的充分条件，并对已发表版本进行证明精简与插图补充。
Spectra of normalized volumes of right-angled hyperbolic polyhedra
本文研究了双曲多面体的标准化体积谱，证明了理想右角双曲多面体的标准化体积谱在特定区间内具有离散和稠密的特性，同时给出了紧凑右角双曲多面体的标准化体积谱的范围和性质。
On lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra in dimensions from 5 to 12
研究通过几何方法确定了5维右角双曲多面体至少有3个理想顶点，7维至少有4个有限顶点，并通过递推关系扩展到12维。
Algebraically fibering group is an algebraic generalization of the fibered 3-manifold group in higher dimensions. Let $M(\mathcal{P})$ and $M(\mathcal{E})$ be the cusped and compact hyperbolic real moment-angled manifolds associated to the hyperbolic right-angled 24-cell $\mathcal{P}$ and the hyperbolic right-angled 120-cell $\mathcal{E}$, respectively. Jankiewicz-Norin-Wise showed in [13] that $π_1(M(\mathcal{P}))$ and $π_1(M(\mathcal{E}))$ are algebraic fibered. Namely, there are two exact sequences $$1\rightarrow H_{\mathcal{P}}\rightarrow π_1(M(\mathcal{P}))\xrightarrow{ϕ_{\mathcal{P}}} \m…
A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension $n=3$ using right-angled dodecahedra and $120$-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $k\geqslant 1$, we build an orientable compact closed $3$-manifold tessellated by $16k$ right-a…
Fixed-point-free involutions on the boundaries of Right-angled Coxeter Groups
本文证明 Nica 关于边界存在无不动点对合的猜想在直角 Coxeter 群情形下的变体，并为原词双曲群问题提供方法启发。
Right-angled Artin groups of large girth and finite volume hyperbolic $3$--manifold groups
本文证明：若Γ是围长至少5的有限单纯图，则有限体积双曲3-流形的基本群不能作为右角Artin群A(Γ)的子群，特别地回答了Reid关于n-圈的问题。
For a compact right-angled polyhedron $R$ in $\mathbb H^3$ denote by $\operatorname{vol} (R)$ the volume and by $\operatorname{vert} (R)$ the number of vertices. Upper and lower bounds for $\operatorname{vol} (R)$ in terms of $\operatorname{vert} (R)$ were obtained in \cite{A09}. Constructing a 2-parameter family of polyhedra, we show that the asymptotic upper bound $5 v_3 / 8$, where $v_3$ is the volume of the ideal regular tetrahedron in $\mathbb H^3$, is a double limit point for ratios $\operatorname{vol} (R) / \operatorname{vert} (R)$. Moreover, we improve the lower bound in the case $\ope…
We prove that among four-dimensional ideal right-angled hyperbolic polytopes the 24-cell is of minimal volume and of minimal facet number. As a corollary, a dimension bound for ideal right-angled hyperbolic polytopes is obtained.
Convex cocompact right-angled Gromov-Thurston polyhedra
针对5维凸余紧双曲反射群的相关问题，本文采用与Lee和Marquis不同的方法，构造了直角情形的对应例子。
For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, we exhibit new geodesics for the arc metric on the Teichm{ü}ller space of an arbitrary surface of negative Euler characteristic with nonempty boundary. We also obtain new geodesics for Thurston's metric on Teichm{ü}ller spaces of hyperbolic surfaces without boundary.
On the finiteness property of hyperbolic simplicial actions: the right-angled Artin groups and their extension graphs
该研究探讨了右角Artin群在扩展图上的作用，证明其渐近平移长度为有理数，并展示了分母可任意变化，同时指出特定条件下渐近平移长度为整数。
We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" graphs and show that every graph $K$ on $\le 8$ vertices either contains one of our examples, or contains a hole of length $\ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a \RAAG to conta…
Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called "flag no-square". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions.
We study the geometry of oriented right-angled hexagons in H^4, the hyperbolic 4-space, via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon so that for the non-augmented sides, we can define quaternion half side-lengths whose angular parts are obtained from half the Euler angles associated to a certain orientation-preserving isometry of the Euclidean 3-space. This generalizes the complex half side-lengths of oriented right-angled hexagons in H^3. We also define appropriate complex half side-lengths for the augmented sides of the hexagon. We further exp…
A threshold for relative hyperbolicity in random right-angled Coxeter groups
本文确定了以Erd{\H o}s--Rényi随机图为表现图的随机右角Coxeter群相对双曲性的阈值为$p=1/\sqrt{n}$，并揭示了其发散性区间及对应随机游走的中心极限定理性质。
We find a sufficient condition for a nerve of a hyperbolic right-angled Coxeter group, under which the boundary of the group is homeomorphic to the Menger curve. We show that this condition is satisfied by many triangulations of surfaces with boundary and other 2-complexes, as well as by some triangulations of disks $D^n$ for arbitrary $n\geq3$.
We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle-free defining graphs such that the associated right-angled Coxeter group has boundary a Menger curve.
We construct a family of right-angled Coxeter groups which provide counter-examples to questions about the stable boundary of a group, one-endedness of quasi-geodesically stable subgroups, and the commensurability types of right-angled Coxeter groups.
By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.
For any right-angled Coxeter group $Γ$ on $k$ generators, we construct proper actions of $Γ$ on $\mathrm{O}(p,q+1)$ by right and left multiplication, and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine transformations, for some $p,q\in\mathbb N$ with $p+q+1=k$. As a consequence, any virtually special group admits proper affine actions on some $\mathbb R^n$: this includes e.g. surface groups, hyperbolic 3-manifold groups, examples of word hyperbolic groups of arbitrarily large virtual cohomological dimension, etc. We also study some examples in cohomological dimension two and four, for which…
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.
We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular polytopes (120-cells and ideal 24-cells). If $M$ is connected, then Vol($W$) < $2^{49}$Vol($M$). This applies for instance to the Borromean link complement. As a consequence, the Borromean link complement bounds geometrically a hyperbolic 4-manifold.
In the present paper we study hyperbolic manifolds that are rational homology 3-spheres obtained by colouring of right--angled polytopes. We study the existence (or absence) of different kinds of symmetries of rational homology spheres that preserve the tessellation of the manifold into polytopes. We also describe how to create colourings with given symmetries.
Character varieties of a transitioning Coxeter 4-orbifold
本文研究 Kerckhoff 和 Storm 的直角 Coxeter 群在双曲、Anti-de Sitter 与半管几何中的特征簇及其坍缩奇点，并利用四维直角尖点群的刚性性质进行分析。
We give a short proof of the following theorem of Sang-hyun Kim: if $A(Γ)$ is a right-angled Artin group with defining graph $Γ$, then $A(Γ)$ contains a hyperbolic surface subgroup if $Γ$ contains an induced subgraph $\bar{C}_n$ for some $n \geq 5$, where $\bar{C}_n$ denotes the complement graph of an $n$-cycle. Furthermore, we give a new proof of Kim's co-contraction theorem.
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes'' studied independently by Haglund and Wise.
We construct an embedding of any right-angled Artin group $G(Δ)$ defined by a graph $Δ$ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of $Δ$. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.
A three-dimensional orthoscheme is defined as a tetrahedron whose base is a right-angled triangle and an edge joining the apex and a non-right-angled vertex is perpendicular to the base. A generalization, called complete orthoschemes, of orthoschemes is known in hyperbolic geometry. Roughly speaking, complete orthoschemes consist of three kinds of polyhedra; either compact, ideal or truncated. We consider a particular family of hyperbolic complete orthoschemes, which share the same base. They are parametrized by the "height", which represents how far the apex is from the base. We prove that th…
Motivated by the notion of cusp excursion in geometrically finite hyperbolic manifolds, we define a notion of excursion in any subgroup of a given group, and study its asymptotic distribution for right-angled Artin groups and graph products. In particular, for any irreducible right-angled Artin group we show that with respect to the counting measure, the maximal excursion of a generic geodesic in any flat tends to $\log n$, where $n$ is the length of the geodesic. In this regard, irreducible RAAGs behave like a free product of groups. In fact, we show that the asymptotic distribution of excurs…
The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to $\mathbb{Z}$ whose kernel has strong topological finiteness proper…
Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex subgroup $H$ of $W$ and any finite subset $\{ γ_1, \ldots , γ_n \} \subset W \setminus H$ there is a surjection $f$ from $W$ to a finite alternating group such that $f (γ_i) \notin f (H)$. A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense. Similarly, finitely gene…
For right-angled Coxeter groups $W_Γ$, we obtain a condition on $Γ$ that is necessary and sufficient to ensure that $W_Γ$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, the peripheral subgroups are both parabolic (in the Coxeter group-theoretic sense) and strongly algebraically thick. We exhibit a polynomial-time algorithm that decides whether a right-angled Coxeter group is thick or relatively hyperbolic. We analyze random graphs in the Erdós-R…
Random infinite ideal angled graphs and ideal hyperbolic polyhedra
本文研究随机无限理想角图与理想双曲多面体的统一化和边界理论，揭示其ICP类型判定及边界一致性，扩展了双曲几何的理论边界。
As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like properties. We prove that these properties are all equivalent and that the contracting boundary is a quasi-isometry invariant. We use this invariant to distinguish the quasi-isometry classes of certain right-angled Coxeter groups.
We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk and the 2--sphere, answering questions due to Crisp--Wiest and M. Kapovich. Another corollary is that a pure braid group contains a closed hyperbolic manifold group as a quasi-isometrically embedded subgroup up to dimension eight. Finally, we show that the isomorphism problem, conj…
Homology growth of polynomially growing mapping tori
本文研究多项式增长自同构的映射圆柱的同调增长性质，证明其具有廉价重建性质并显示torsion同调增长在Farber序列下消失。
We consider two manifestations of non-positive curvature: acylindrical actions on hyperbolic spaces and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for studying many important families of groups, including mapping class groups, right-angled Coxeter and Artin groups, most 3-manifold groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces, so it is natural to search for a "best" one. The set of all cobounded acylindric…
We provide, for hyperbolic and flat 3-manifolds, obstructions to bounding hyperbolic 4-manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.
The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of …
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