Growth rates of small cancellation groups

关于 Gromov 的一个小消去定理

On a small cancellation theorem of Gromov

本文通过组合方法证明了 Gromov 的定理，将小消去理论扩展至源自标记图的群展示，拓宽了其适用范围。

Generalized Small Cancellation Presentations for Automatic Groups

By a result of Gersten and Short finite presentations satisfying the usual non-metric small cancellation conditions present biautomatic groups. We show that in the case in which all pieces have length one, a generalization of the C(3)-T(6) condition yields a larger collection of biautomatic groups.

Small cancellation theory and Burnside problem

In these notes we detail the geometrical approach of small cancellation theory used by T. Delzant and M. Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.

Cubulating Small Cancellation Free Products

We give a simplified approach to the cubulation of small-cancellation quotients of free products of cubulated groups. We construct fundamental groups of compact nonpositively curved cube complexes that do not virtually split.

关于图形C(4)-T(4)和C(6)小消去复形的局部到全局不动点性质

Local-to-global fixed point properties for graphical C(4)-T(4) and C(6) small cancellation complexes

研究由图形小消去表示定义的群，证明C(4)-T(4)表示定义的群的挠子群有限，在特定假设下C(6)表示定义的群也如此，通过关于挠群作用在单连通图形小消去复形上的局部到全局不动点性质得出结论。

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…

Infinitely presented graphical small cancellation groups are acylindrically hyperbolic

We prove that infinitely presented graphical $Gr(7)$ small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical $C(7)$-groups and, hence, classical $C'(\frac{1}{6})$-groups are acylindrically hyperbolic. We also prove the analogous statements for the larger class of graphical small cancellation presentations over free products. We construct infinitely presented classical $C'(\frac{1}{6})$-groups that provide new examples of divergence functions of groups.

A new proof for small cancellation conditions of 2-bridge link groups

In this paper, we give a simple proof for the small cancellation conditions of the upper presentations of 2-bridge link groups, which holds the key to the proof of the main result of [1]. We also give an alternative proof of the main result of [1] using transfinite induction.

A sample iterated small cancellation theory for groups of Burnside type

We develop yet another technique to present the free Burnside group $B(m,n)$ of odd exponent $n$ with $m\ge2$ generators as a group satisfying a certain iterated small cancellation condition. Using the approach, we provide a reasonably accessible proof that $B(m,n)$ is infinite with a moderate bound $n > 2000$ on the odd exponent $n$.

Some small cancellation properties of random groups

We work in the density model of random groups. We prove that they satisfy an isoperimetric inequality with sharp constant $1-2d$ depending upon the density parameter $d$. This implies in particular a property generalizing the ordinary $C'$ small cancellation condition, which could be termed ``macroscopic small cancellation''. This also sharpens the evaluation of the hyperbolicity constant $δ$. As a consequence we get that the standard presentation of a random group at density $d<1/5$ satisfies the Dehn algorithm and Greendlinger's Lemma, and that it does not for $d>1/5$.

具有和不具有σ-紧Morse边界的小消去群

Small cancellation groups with and without sigma-compact Morse boundary

该研究构造了首批具非σ-紧Morse边界的经典小消去群实例，完整刻画了$C'(1/6)$-群Morse边界的σ-紧性条件，证明该性质可区分小消去群的拟等距类型，还给出了所有Morse射线均强收缩的$C'(1/6)$-群实例。

Acylindrical hyperbolicity of cubical small-cancellation groups

We provide an analogue of Strebel's classification of geodesic triangles in classical $C'(\frac16)$ groups for groups given by Wise's cubical presentations satisfying sufficiently strong metric cubical small cancellation conditions. Using our classification, we prove that, except in specific degenerate cases, such groups are acylindrically hyperbolic.

Commutators cannot be proper powers in metric small-cancellation torsion-free groups

A nontrivial commutator cannot be a proper power in a torsion-free group satisfying C'(λ) small cancellation condition with sufficiently small λ.

Graphical small cancellation groups with the Haagerup property

We prove the Haagerup property (= Gromov's a-T-menability) for finitely generated groups defined by infinite presentations satisfying the graphical C'(lambda)-small cancellation condition with respect to graphs endowed with a compatible wall structure. We deduce that these groups are coarsely embeddable into a Hilbert space and that the strong Baum-Connes conjecture and, hence, the Baum-Connes conjecture with arbitrary coefficients hold for them. As the main step we show that C'(lambda)-complexes satisfy the linear separation property. Our result provides many new examples and a general techni…

非度量立方小消去中的双曲性

Hyperbolicity in non-metric cubical small-cancellation

本文研究非正弯曲立方复形基本群的商群双曲性，证明在充分非度量立方小消去条件下，双曲基本群的相应商仍双曲，并推广经典 $C(7)$ 小消去群结论。

小消去群是双精确的

Small cancellation groups are bi-exact

本文证明了有限生成的$C'(\ rac{1}{33})$-群（不必有限表示）是双精确的，从而扩展了双精确群的已知类。

Generalized Small Cancellation Theory

We present four generalized small cancellation conditions for finite presentations and solve the word- and conjugacy problem in each case. Our conditions $W$ and $W^*$ contain the non-metric small cancellation cases C(6), C(4)T(4), C(3)T(6) (see [LS]) but are considerably more general. $W$ also contains as a special case the small cancellation condition $W(6)$ of Juhasz [J2]. If a finite presentation satisfies $W$ or $W^*$ then it has a quadratic isoperimetric inequality and therefore solvable word problem. For the class $W$ this was first observed by Gersten in [G7] which also contains an ide…

通过图小消去构造 $FP$ 型群

Groups of type $FP$ via graphical small cancellation

本文利用 Gromov 图小消去理论，而非 Morse 理论，构造了不可数多个 $FP$ 型群，提供了新的构造方法。

Small cancellations over relatively hyperbolic groups and embedding theorems

We generalize the small cancellation theory over hyperbolic groups developed by Olshanskii to the case of relatively hyperbolic groups. This allows us to construct infinite finitely generated groups with exactly $n$ conjugacy classes for every $n\ge 2$. In particular, we give the affirmative answer to the well--known question of the existence of a finitely generated group $G$ other than $\mathbb Z/2\mathbb Z$ such that all nontrivial elements of $G$ are conjugate.

非乘积的立方小消去商

Cubical small-cancellation quotients of non-products

证明了紧非正曲率非乘积立方复形的基本群存在保持相同性质的立方复形基本群作为其真商。

Local Quasiconvexity of Groups acting on Small Cancellation Complexes

Given a group acting cellularly and cocompactly on a simply-connected 2-complex, we provide a criterion establishing that all finitely generated subgroups have quasiconvex orbits. This work generalizes the "perimeter method". As an application, we show that high-powered one-relator products $A \ast B / \nclose{r^n}$ are coherent if $A$ and $B$ are coherent.

Relative Hyperbolicity of Graphical Small Cancellation Groups

A piece of a labelled graph $Γ$ defined by D. Gruber is a labelled path that embeds into $Γ$ in two essentially different ways. We prove that graphical $Gr'(\frac{1}{6})$ small cancellation groups whose associated pieces have uniformly bounded length are relative hyperbolic. In fact, we show that the Cayley graph of such group presentation is asymptotically tree-graded with respect to the collection of all embedded components of the defining graph $Γ$, if and only if the pieces of $Γ$ are uniformly bounded. This implies the relative hyperbolicity by a result of C. Druţu, D. Osin and M. Sapir.

Actions of small cancellation groups on hyperbolic spaces

We generalize Gruber--Sisto's construction of the coned--off graph of a small cancellation group to build a partially ordered set $\mathcal{TC}$ of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber--Sisto coned--off graph. In almost all cases $\mathcal{TC}$ is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions $[G\curvearrowright X] \preceq [G\curvearrowright Y]$ in this poset, there is an embeddeding $ι:P(ω)\to\mathcal{TC}$ such that $ι(\emptyset)=[G\curvearrowr…

小取消群的扭子群

Torsion subgroups of small cancellation groups

该研究证明了特定小取消群的扭子群为有限循环群，并探讨了局部椭圆作用和自动连续性相关结论。

Rips-Segev torsion-free groups without the unique product property

We generalize the graphical small cancellation theory of Gromov to a graphical small cancellation theory over the free product. We extend Gromov's small cancellation theorem to the free product. We explain and generalize Rips-Segev's construction of torsion-free groups without the unique product property by viewing these groups as given by graphical small cancellation presentations over the free product. Our graphical small cancellation theorem then provides first examples of Gromov hyperbolic groups without the unique product property. We construct uncountably many non-isomorphic torsion-free…

无扭双曲群中的一致小消去理论

Uniform small cancellation in torsion-free hyperbolic groups

该研究针对非初等无扭双曲群建立一致小消去定理，并将其应用于证明这类群的一阶理论不具有模型伴侣。

CAT(-1) metrics on small cancellation groups

We give a proof that groups satisfying the "uniform C'(1/6)" small cancellation condition admit a geometric action on a CAT(-1) space. It follows that random groups at density <1/12 are CAT(-1). The proof consists of a direct construction of a piecewise hyperbolic structure on the presentation complex of such a group, together with folding moves to make the complex negatively curved. The argument was originally suggested by Gromov.

Asphericity and small cancellation theory for rotation family of groups

Using small cancellation for rotating families of groups, we construct new examples of aspherical polyhedra.

Small cancellation groups and translation numbers

In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation dis crete in the strongest possible sense and that in these groups for any $g$ and any $n$ there is an algorithm deciding whether or not the equation $ x^n=g$ has a solution. There is also an algorithm for calculating for each $g$ the maximum $n$ such that $g$ is an $n$-th power of some element. We also note that these groups cannot contain isomorphic copies of the gr oup of $p$-adic fractions and so in particular of the group of rational numbers. Besides we show that for C''(4)-T(4) and C…

Small cancellation theory over Burnside groups

We develop a version of small cancellation theory in the variety of Burnside groups. More precisely, we show that there exists a critical exponent $n_0$ such that for every odd integer $n\geq n_0$, the well-known classical $C'(1/6)$-small cancellation theory, as well as its graphical generalization and its version for free products, produce examples of infinite $n$-periodic groups. Our result gives a powerful tool for producing (uncountable collections of) examples of $n$-periodic groups with prescribed properties. It can be applied without any prior knowledge in the subject of $n$-periodic gr…

Introduction to a small cancellation theorem

This note is intended as an introduction to two previous works respectively by Dahmani, Guirardel, Osin, and by Cantat, Lamy. We give two proofs of a Small Cancellation Theorem for groups acting on a simplicial tree. We discuss the application to the group of plane polynomial automorphisms over any ground field.

图形小消去与边界作用的超有限性

Graphical small cancellation and hyperfiniteness of boundary actions

研究（无限表示的）图形小消去群在锥化Cayley图的Gromov边界上的作用，证明包括（无限表示的）经典小消去群在内的一类图形小消去群具有超有限边界作用。

Algebraic C(4) and T(4) groups are bi-automatic

S. Gersten and H. Short have proved that if a group has a presentation which satisfies the algebraic C(4) and T(4) small-cancellation condition then the group is automatic. Their proof contains a gap which we aim to close. To do that we distinguish between algebraic small-cancellation conditions and geometric small cancellation conditions (which are conditions on the van Kampen diagrams). We show that, under certain additional requirements, geometric C(4) and T(4) small-cancellation conditions imply bi-automaticity. The additional requirements include a restriction on the labels of edges in mi…

Complexes of groups and geometric small cancellation over graphs of groups

We explain and generalise a construction due to Gromov to realise geometric small cancellation groups over graphs of groups as fundamental groups of non-positively curved 2-dimensional complexes of groups. We then give conditions so that the hyperbolicity and some finiteness properties of the small cancellation quotient can be deduced from analogous properties for the local groups of the initial graph of groups.

非度量小消去中的Cohen--Lyndon性质

The Cohen--Lyndon property in non-metric small-cancellation

本文证明非度量小消去商群满足Cohen--Lyndon性质，推广了度量情形结果，并解答了Lyndon与Wall的长期问题。

小取消条件与具有 (T) 性质的超限群在外自同构群中的作用

Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces

该研究证明了每个有限群都可以作为具有 (T) 性质的超限群的外自同构群，并展示了如何构造具有特定外自同构群的超限群。

图论C(3)-T(6)蕴含CAT(0)

Graphical C(3)-T(6) implies CAT(0)

本文通过图论C(3)-T(6)复形构造局部CAT(0)度量，扩展了经典小取消理论的应用范围。

小取消平坦环面定理

The small cancellation flat torus theorem

本文研究了满足C(6)、C(4)-T(4)和C(3)-T(6)条件的小取消复形的平坦环面定理，探讨了这些复形的几何性质及与CAT(0)空间的关系。

The Burnside groups and small cancellation theory

In a pair of recent papers (one to appear and one forthcoming), the author develops a general version of small cancellation theory applicable in higher dimensions, and then applies this theory to the Burnside groups of sufficiently large exponent. The present article gives a brief introduction to the methods and techniques involved in the proofs. Many of the ideas originate with the recent work of A. Yu. Olshanskii and S. V. Ivanov. Some familiarity with their work on the Burnside problem is assumed.

随机分支覆盖的有限可接受2复形

Small Cancellation for Random Branched Covers of Groups

本文研究了有限可接受2复形的随机分支覆盖，证明其几乎必然满足几何小取消条件，并具有超双曲群性质。

Infinitely presented small cancellation groups have the Haagerup property

We prove the Haagerup property (= Gromov's a-T-menability) for finitely generated groups defined by infinite presentations satisfying the C'(1/6)-small cancellation condition. We deduce that these groups are coarsely embeddable into a Hilbert space and that the strong Baum-Connes conjecture holds for them. The result is a first non-trivial advancement in understanding groups with such properties among infinitely presented non-amenable direct limits of hyperbolic groups. The proof uses the structure of a space with walls introduced by Wise. As the main step we show that C'(1/6)-complexes satisf…

A combination theorem for cubulation in small cancellation theory over free products

We prove that a group obtained as a quotient of the free product of finitely many cubulable groups by a finite set of relators satisfying the classical $C'(1/6)$--small cancellation condition is cubulable. This yields a new large class of relatively hyperbolic groups that can be cubulated, and constitutes the first instance of a cubulability theorem for relatively hyperbolic groups which does not require any geometric assumption on the peripheral subgroups besides their cubulability. We do this by constructing appropriate wallspace structures for such groups, by combining walls of the free fac…

Small cancellation labellings of some infinite graphs and applications

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: - We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu's Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. - We construct the first examples of finitely generated groups, with expanders …

Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

We present a metric condition ${\LARGEτ}'$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition ${\LARGEτ}'$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, ${\LARGEτ}'$-groups have quadratic Dehn functions and solvable conjugacy problem. In the s…

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ …

Small cancelation rings

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out…

小消去自由积的相对立方化

Relative Cubulation of Small Cancellation Free Products

该研究扩展了在CAT(0)立方复形上有相对几何作用的群类，证明其在C'(1/6)-小消去自由积下封闭，还得到剩余有限群的这类自由积仍为剩余有限的新应用。

环的类群小消去理论

Group-like small cancellation theory for rings

本文将群论小消去理论推广到具有可逆元基的结合代数，通过三条公理研究自由群群代数的商环，证明其非平凡性并给出基、滤链与结构定理，同时建立 Gröbner 基与理想成员问题的贪心算法。

图小消去理论中的优化

Optimization in graphical small cancellation theory

该研究聚焦图小消去理论，简化Osajda的概率方法，在构造含扩张图族的Gromov怪物群时显著减少生成元数量，推进相关群构造的优化。

Nielsen equivalence in small cancellation groups

Let $G$ be a group given by the presentation [<a_1,...,a_k,b_1,... b_k\,| a_i=u_i(\bar b), b_i=v_i(\bar a) \hbox{for} 1\le i\le k>,] where $k\ge 2$ and where the $u_i\in F(b_1,..., b_k)$ and $w_i\in F(a_1,..., a_k)$ are random words. Generically such a group is a small cancellation group and it is clear that $(a_1,...,a_k)$ and $(b_1,...,b_k)$ are generating $n$-tuples for $G$. We prove that for generic choices of $u_1,..., u_k$ and $v_1,..., v_k$ the "once-stabilized" tuples $(a_1,..., a_k,1)$ and $(b_1,...,b_k,1)$ are not Nielsen equivalent in $G$. This provides a counter-example for a Wiego…

Conformal dimension via subcomplexes for small cancellation and random groups

We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like l^K in the length l of the relators, then a.a.s. such a random group has conformal dimension 2+K+o(1). In Gromov's density model, a random group at density d<1/8 a.a.s. has conformal dimension $\asymp dl / |\log d|$. The upper bound for C'(1/8) groups has two main ingredients: $\ell_p$-cohomology (following Bourdon-Kleiner), and walls in the Cayley co…

Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power…

Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface: improving the bound from exponential to at most quadratic in the complexity of the surface. We also apply the main result to various other hierarchically hyperbolic groups and spaces. We also prove a small-cancellation result namely: if $G$ is a hierarchically hyperbolic group, $H\leq G$ is a suitable hyperbolically em…

线性系统与小取消的算子解

Operator solutions of linear systems and small cancellation

研究证明，当图的最小顶点度至少为d且圈长至少为g时，图的 incidence 系统在Z_p上存在量子解。特别地，存在某些线性系统，其对应的非本地游戏有完美算子策略但无完美经典策略。