A note on quotients of word hyperbolic groups with Property (T)

Property (T) in random quotients of hyperbolic groups at densities above 1/3

We study random quotients of a fixed non-elementary hyperbolic group in the Gromov density model. Let $G=\langle S\;\vert\; T\rangle $ be a finite presentation of a non-elementary hyperbolic group, and let $Ann_{l,ω}(G)$ be the set of elements of norm between $l-ω(l)$ and $l$ in $G$. A random quotient at density $d$ and length $ω$-near $l$ is defined by killing a uniformly randomly chosen set of $\vert S_{l}(G)\vert ^{d}$ words in $Ann_{l,ω(l)}(G)$, where $ω(l) =o_{l}(l)$. We prove that for any d>1/3, such a quotient has Property (T) with probability tending to $1$ as $l$ tends to infinity. Th…

Periodic quotients of hyperbolic and large groups

Let $G$ be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this paper we describe several approaches for constructing continuous families of periodic quotients of $G$ with various properties. The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of $G$. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer $n$ such that $G/G^n$ is an infinite group of exponent $n$. The fourth approach start…

商饱和群

Quotient-saturated groups

本文提出商饱和概念以刻画群的商结构规模，给出有限展示群商饱和的充分条件，并证明双曲群及柱面双曲群的相关非初等有限展示子群具有该性质。

由有限商决定的双曲自由循环群

A hyperbolic free-by-cyclic group determined by its finite quotients

本文证明了一个具体的双曲自由循环群在自由循环群类中是 profinitely 刚性的，即由其有限商唯一决定，这是此类群的第一个例子。

有界上同调、商扩张与分层双曲性

Bounded cohomology, quotient extensions, and hierarchical hyperbolicity

研究有界中心扩张相关问题，证明HHG的有界中心扩张仍是HHG及相关逆命题，探讨有界中心扩张商的有界性问题等价于拟同态可扩展性问题，还表明4股辫群特定商是HHG及有界中心扩张，并对扩展结果至所有映射类群进行推测。

层级双曲性的组合视角及其在映射类群商群中的应用

A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups

本研究提出基于双曲单纯复形作用的层级双曲群组合判据，证明映射类群模Dehn扭大幂次的商群为层级双曲群，构造非初等双曲商群并将相关有限商问题归约为特定双曲群的剩余有限性问题。

Word Hyperbolic Semigroups

The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition. Roughly speaking a semigroup is word hyperbolic if its multiplication table is a context free language. For groups this definition is equivalent to the original geometric one. We also briefly consider word problems of semigroups.

On stable norm in word hyperbolic groups

This work is concerned with the stable norm in word hyperbolic groups as defined by Gromov. We give a short elementary proof of one of its basic property, that is existence of a computable uniform non null lower bound for stable norm in a word hyperbolic group.

The compressed word problem in relatively hyperbolic groups

We prove that the compressed word problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.

On the Definition of Word Hyperbolic Groups

Formal languages based on the multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups.

Central Extensions of Word Hyperbolic Groups

Thurston has claimed (unpublished) that central extensions of word hyperbolic groups by finitely generated abelian groups are automatic. We show that they are in fact biautomatic. Further, we show that every 2-dimensional cohomology class on a word hyperbolic group can be represented by a bounded 2-cocycle. This lends weight to the claim of Gromov that for a word hyperbolic group, the cohomology in every dimension is bounded.

On Residual Properties of Word Hyperbolic Groups

For a fixed word hyperbolic group we compare different residual properties related to quasiconvex subgroups.

Context-free rewriting systems and word-hyperbolic structures with uniqueness

This paper proves that any monoid presented by a confluent context-free monadic rewriting system is word-hyperbolic. This result then applied to answer a question asked by Duncan & Gilman by exhibiting an example of a word-hyperbolic monoid that does not admit a word-hyperbolic structure with uniqueness (that is, in which the language of representatives maps bijectively onto the monoid).

短层次超几何群II：商和Artin群的Hopf性质

Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups

本文证明大多数大类型和超几何类型的Artin群是Hopf群，并展示大部分五孔球面映射群的商是层次超几何的。

随机商保持圆柱形和层次超几何性

Random quotients preserve acylindrical and hierarchical hyperbolicity

本文提出了一种随机商模型，证明了随机商保持圆柱形和层次超几何性，并展示了其在超几何群中的应用。

Hyperbolic quotients of projection complexes

This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions that the quotient complex is $δ$-hyperbolic. Additionally, under certain circumstan…

Fixed-point property of random quotients by plain words

We show a fixed-point property of random quotients by plain words for a wide class of CAT(0) spaces.

关于WPD作用下双曲空间的随机商及边界的线性连通性

Random Quotients of Hyperbolic Spaces with Respect to WPD Actions and Linear Connectivity of the Boundary

该研究探讨两类双曲群的随机商，证明 acylindrically 双曲群在满足特定作用条件时，其随机元素正规闭包的商空间边界线性连通，分层双曲群对应商空间边界同胚于 Nöbeling 曲线。

立方化双曲群的随机商群的立方化

Cubulating random quotients of hyperbolic cubulated groups

本文证明立方化双曲群的低密度随机商群仍为立方化且双曲，利用立方小消去理论、共轭类指数增长及超平面稳定子增长较慢等要素。

短且高度非本原的字生成双曲单关系群

Short, highly imprimitive words yield hyperbolic one-relator groups

本文为 Louder 和 Wilton 的猜想提供了实验支持，验证了非本原秩大于二的字能生成双曲单关系群。

The generalised word problem in hyperbolic and relatively hyperbolic groups

We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller-Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic gro…

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。

An example of a non-quasiconvex subgroup of a word hyperbolic group with exotic limit set

We construct an example of a torsion free freely indecomposable finitely presented non-quasiconvex subgroup $H$ of a word hyperbolic group $G$ such that the limit set of $H$ is not the limit set of a quasiconvex subgroup of $G$. In particular, this gives a counterexample to the conjecture of G.Swarup that a finitely presented one-ended subgroup of a word hyperbolic group is quasiconvex if and only if it has finite index in its virtual normalizer.

层级双曲性、球商与光滑三次曲面模空间

Hierarchical hyperbolicity, ball quotients, and the moduli space of smooth cubic surfaces

本文证明光滑复三次曲面模空间的轨道基本群是层级双曲群，利用不完备复双曲度量，并推广至其他模空间。

The generalised word problem for subgroups of hyperbolic groups

We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of infinite index with context-free generalised word problem, and that, for any hyperbolic group, the generalised word problem of a torsion-free quasiconvex subgroup is recognised by a real-time Turing machine.

Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups

For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.

任意环上2维字双曲群的子群

Subgroups of word hyperbolic groups in dimension 2 over arbitrary rings

本文利用任意环上的等周不等式将Gersten关于2维字双曲群子群的双曲性结果推广至任意环，并给出了高阶双曲群的子群保持同阶双曲性的条件及组合定义。

Sharp phase transition theorems for hyperbolicity of random groups

We prove that in various natural models of a random quotient of a group, depending on a density parameter, for each hyperbolic group there is some critical density under which a random quotient is still hyperbolic with high probability, whereas above this critical value a random quotient is very probably trivial. We give explicit characterizations of these critical densities for the various models.

A series of word-hyperbolic Coxeter groups

For each positive integer $k$ we present an example of Coxeter system $(G_k,S_k)$ such that $G_k$ is a word-hyperbolic Coxeter group, for any two generating reflections $s,t\in S_k$ the product $st$ has finite order, and the Coxeter graph of $S_k$ has negative inertia index equal to $k$. In particular, $G_k$ cannot be embedded into pseudo-orthogonal group $O(n,k-1)$ for any $n$ as a reflection group with generating set being reflections w.r.t. $(G_k,S_k)$

Subgroups of word hyperbolic groups in rational dimension 2

A result of Gersten states that if $G$ is a hyperbolic group with integral cohomological dimension $\mathsf{cd}_{\mathbb{Z}}(G)=2$ then every finitely presented subgroup is hyperbolic. We generalize this result for the rational case $\mathsf{cd}_{\mathbb{Q}}(G)=2$. In particular, our result applies to the class of torsion-free hyperbolic groups $G$ with $\mathsf{cd}_{\mathbb{Z}}(G)=3$ and $\mathsf{cd}_{\mathbb{Q}}(G)=2$ discovered by Bestvina and Mess.

Hyperbolic groups and free constructions

We investigate which free constructions (amalgamated products and HNN-extensions) over word hyperbolic groups produce groups that are again word hyperbolic. A complete answer is obtained for the case when the amalgamated subgroups are virtually cyclic. The results are applied, in particular, to show that a ${\bf Q}$-completion of a torsion-free hyperbolic group has solvable word problem and conjugacy problem.

Computability of finite quotients of finitely generated groups

We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can be even not recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem while still not…

群论Dehn填充的上同调 II

Cohomology of group theoretic Dehn fillings II

本文研究群论Dehn填充的上同调，利用Cohen-Lyndon性质推导谱序列计算商群上同调，建立相对上同调同构并推广相关结果，还强化了柱形双曲群的SQ-泛性等结论。

A non-quasiconvexity embedding theorem for hyperbolic groups

We show that if $G$ is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group $G^*$, such that $G$ is a subgroup of $G^*$ but $G$ is not quasiconvex in $G^*$.

Stable commutator length in word-hyperbolic groups

In this paper we obtain uniform positive lower bounds on stable commutator length in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on the Bass-Serre tree). If G is a word hyperbolic group which is delta hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on delta and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length is at least equal to …

Decision problems for word-hyperbolic semigroups

This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These…

映射到超几何群积上的满射不可判定性

Undecidability of epimorphisms onto products of hyperbolic groups

研究证明了某些超几何群积的满射不可判定性，通过构造递归序列的超几何群来展示算法无法判断商群同构性。

One Relator Quotients of Graph Products

In this paper, we generalise Magnus' Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.

Simple closed curves, word length, and nilpotent quotients of free groups

We consider the fundamental group $π$ of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of $π$ has finite diameter with respect to the word metric given by this set. This is in contrast with a result of Danny Calegari that shows that $π$ has infinite diameter with respect to this set. Furthermore, we give a general criterion for a finitely generated group equipped with a generating set to have this property.

Computation in word-hyperbolic groups

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.

Torsion-free Word Hyperbolic Groups are Noncommutatively Slender

In this note we prove the claim given in the title. A group G is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to G factors through projection to a canonical free subgroup. Graham Higman, in his seminal 1952 paper, proved that free groups are noncommutatively slender. Such groups were first defined by K. Eda. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few- relator sense of Gromov are noncommutatively slender.

Hierarchical hyperbolicity of hyperbolic-2-decomposable groups

Let G be a graph of hyperbolic groups with 2-ended edge groups. We show that G is hierarchically hyperbolic if and only if G has no distorted infinite cyclic subgroup. More precisely, we show that G is hierarchically hyperbolic if and only if G does not contain certain quotients of Baumslag-Solitar groups. As a consequence, we obtain several new results about this class, such as quadratic isoperimetric inequality and finite asymptotic dimension.

一个非一致指数增长的圆柱双曲群

An acylindrically hyperbolic group of non-uniform exponential growth

构造了一个具有非一致指数增长和Kazhdan性质(T)的圆柱双曲群，该群是递增有限秩自由群的自同构群的公共商。

The Cohomology of canonical quotients of free groups and Lyndon words

For a prime number $p$ and a free profinite group $S$, let $S^{(n,p)}$ be the $n$th term of its lower $p$-central filtration, and $S^{[n,p]}$ the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group $H^2(S^{[n,p]},\mathbb{Z}/p)$, which we call the Lyndon basis, and use it to obtain structural results on this group. We show a duality between the Lyndon basis and canonical generators of $S^{(n,p)}/S^{(n+1,p)}$. We prove that the cohomology group satisfies shuffle relations, which for small values of $n$ fully describe it.

Engulfing in word-hyperbolic groups

We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.

一种广义的无限词幺半群：渐近前后缀商与代数结构

A generalized monoid of infinite words: asymptotic prefix-suffix quotients and algebraic structure

本研究构造了基于双向前后缀度量渐近等价的广义词幺半群\\(\tilde{\Sigma}^*\\)，揭示其代数性质，证明它严格扩张经典无限词集合，并建立了对数前缀泛函的良定义作用。

Multiplication tables and word-hyperbolicity in free products of semigroups, monoids, and groups

This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied…

Boundaries of hyperbolic groups

We survey the known results regarding the boundaries of word-hyperbolic groups.

A word-hyperbolic special monoid with undecidable Diophantine problem

The Diophantine problem for a monoid $M$ is the decision problem to decide whether any given system of equations has a solution in $M$. In this note, we give a simple example of a context-free, word-hyperbolic, finitely presented, special monoid $M$ with trivial group of units, and such that the Diophantine problem is undecidable in $M$. This answers two questions asked by Garreta & Gray in 2019, and shows that the decidability of the Diophantine problem in hyperbolic groups, as proved by Dahmani & Guirardel, does not generalise to word-hyperbolic monoids.

Linearity and indiscreteness of amalgamated products of hyperbolic groups

We discuss the linearity and discreteness of amalgamated products of linear word-hyperbolic groups. In particular, we prove that the double of an Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups which do not admit any discrete and faithful representation in rank 1. We also build new examples of non-linear word-hyperbolic groups, elaborating on a previous work of Canary--Stover--Tsouvalas.

Hyperbolic 3-Manifolds Groups are Subgroup Conjugacy Separable

A group $G$ is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of $G$, there exists a finite quotient of $G$ where the images of these subgroups are not conjugate. It is proved that the fundamental group of a hyperbolic 3-manifold (closed or with cusps) is subgroup conjugacy separable.

Algorithmic properties of inverse monoids with hyperbolic and tree-like Schützenberger graphs

We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.

Small cancellation in acylindrically hyperbolic groups

We generalize a version of small cancellation theory to the class of acylindrically hyperbolic groups. This class contains many groups which admit some natural action on a hyperbolic space, including non-elementary hyperbolic and relatively hyperbolic groups, mapping class groups, and groups of outer automorphisms of free groups. Several applications of this small cancellation theory are given, including to Frattini subgroups and Kazhdan constants, the construction of various \exotic"' quotients, and to approximating acylindrically hyperbolic groups in the topology of marked group presentation…

字双曲群的非噪声敏感性

Non-noise sensitivity for word hyperbolic groups

该研究针对字双曲群上的非初等随机游走，证明其在小噪声参数、有限一阶矩条件下具备强非噪声敏感性，拓展了群论与随机过程的相关结论。