Asymptotic dimension of one relator groups
Dmitry Matsnev
我们证明了一个关系子群被看作是度量空间关于字长度量有有限的渐近维数在Gromov意义下，并给出了其渐近维数的估计关系子长度。
We show that one relator groups viewed as metric spaces with respect to the word-length metric have finite asymptotic dimension in the sense of Gromov and give an estimate of their asymptotic dimension in terms of the relator length.
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Asymptotic Dimension of Graphs of Groups and One Relator Groups
本文证明HNN-扩张渐近维数的新不等式，证实单关系子群渐近维数至多为2的猜想，并计算右角Artin群精确维数及给出图的基本群的新上界。
Delzant's T-invariant, Kolmogorov complexity and one-relator groups
我们证明，“几乎一般”的一个关系组Delzant的$T$不变（它的措施的最小尺寸的一个有限的介绍一组）是可比的大小与长度的定义关系。证明依赖于我们以前的结果同构刚性的一般单关系群和Kolmogorov-Chaitin复杂性理论的方法。我们还给出了一个精确的渐近估计（当$k$是固定的和$n$走向无穷大）的数目$I_{k,n}$的同构类的$k$ -生成器单关系员组与循环减少定义关系员的长度$n$：\[ I_{k,n}\sim \frac{(2k-1)^n}{nk!2^{k+1}}. \]这里$f(n)\sim g(n)$的意思是$\lim_{n\to\infty} f(n)/g(n)=1$。
Gromov's Conjecture for Product of Baumslag-Solitar groups and some other One-relator groups
证明Baumslag-Solitar群及部分推广不可定向曲面群的单关系群是虚拟2-可避免的，结合前期结果，证实了基本群为这些群乘积的闭定向自旋流形的Gromov PSC流形万有覆盖宏观维数猜想。
For every one-relator monoid $M = \langle A \mid u=v \rangle$ with $u, v \in A^*$ we construct a contractible $M$-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type ${\rm FP}_\infty$, answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most $2$, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly g…
How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group $G$ is said to be properly 3-realizable if there exists a compact 2-polyhedron $K$ with $π_1(K) \cong G$ whose universal cover $\tilde{K}$ has the proper homotopy type of a PL 3-manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated one-relator groups and show that those having finitely many ends are properly 3-realizable, by describing what the fundamental pro-group l…
On the coherence of one-relator groups and their group algebras
本文证明了单关系群的凝聚性，解决了 Gilbert Baumslag 的著名问题，并将该策略应用于上同调维数为2的群类，证明了二维复形基本群、交错展示群、部分 Coxeter 群及其群代数的凝聚性。
Irreducible components in an algebraic variety of representations of a family of one-relator groups
给定一个有限生成群G，集合Hom（G，SL_2 C）继承一个代数簇R（G）的结构，R（G）称为G的“表示簇”.这个代数簇是G的一个不变量。令G_{pt} =< a，b; a^p= b^t>，其中p、t是大于1的整数。本文给出了仿射代数簇R（G_{pt}）的四维不可约分支数的一个公式。本文主要定理的一个直接结果是：若K是环面纽结，则它的亏格等于R（G_{pt}）中对应于其纽结群G_{pt}的四维分支的个数.
L^2-Betti numbers of one-relator groups
我们确定了所有单关系群和所有曲面加一关系群（曲面加一关系群由Hempel引入，他称之为单关系曲面群）的L^2-贝蒂数.特别地，我们证明了对所有这样的群G，L^2-贝蒂数b_n^{（2）}（G）对所有n>1都为0。我们还得到了关于左序群的L^2-上同调的一些信息，并推出了非L^2的结果：在同调维数为1的左序群中，所有的双生成子群都是自由的。
We give a lower and an upper bound for the conformal dimension of the boundaries of certain small cancellation groups. We apply these bounds to the few relator and density models for random groups. This gives generic bounds of the following form, where $l$ is the relator length, going to infinity. (a) $1 + 1/C < \Cdim(\bdry G) < C l / \log(l)$, for the few relator model, and (b) $1 + l / (C\log(l)) < \Cdim(\bdry G) < C l$, for the density model, at densities $d < 1/16$. In particular, for the density model at densities $d < 1/16$, as the relator length $l$ goes to infinity, the random groups w…
Two-sided homological properties of special and one-relator monoids
研究特殊和一关系单调群的双侧同调有限性性质，证明其与单位群的同调有限性关系，并展示一关系单调群的Hochschild上同度量与其单位群的上同度量的关系，以及一关系单调群的双侧同调有限性。
The surjectivity problem for one-generator,, one-relator extensions of torsion-free groups
本文利用Klyachko的方法证明了G到G-hat的自然映射是满射的，其中G是一个无挠的群，G-hat是通过增加一个新的生成元t和一个新的关系子w得到的，仅当对G中的某个g，w与gt或gt^{-1}共轭.这解决了一个特殊情况下的满射性问题的群扩展，提出了科恩[怀特黑德扭转，群扩展，塞曼的猜想在高维，拓扑学，16（1977）79- 88]。
Using stably free non-free relation modules we construct an infinite collection of 2-dimensional homotopy types, each of Euler-characteristic one and with trefoil fundamental group. This provides an affirmative answer to a question asked by Berridge and Dunwoody [J. London Math. Soc. 19 (1979) 433-436]. We also give new examples of exotic relation modules. We show that the relation module associated with the generating set x, y^4 for the Baumslag-Solitar group <x, y | xy^2x^{-1}=y^3> is stably free non-free of rank one.
Random quotients of the modular group are rigid and essentially incompressible
证明了对任意正整数$m\ge 1$，模群$M = PSL(2,\mathbb{Z})$的$m$ -关系子一般满足一个很强的Mostow型\emph{isomorphism rigidity} .我们还证明了，这样的连续性是一般的“本质上不可压缩”。通过这一点，我们的意思是，他们的“绝对$T$不变”，衡量最小的大小，任何可能的有限介绍的组，是有界的功能，这是几乎线性的长度方面的给定介绍。我们计算了$m$的\emph{isomorphism types}的$I_m(n)$-$M$的关系子的精确渐近性，其中所有定义的关系子都是$M$中长度为$n$的循环约简字。我们得到其他的代数结果，并表明，这样的群是完全的，霍普夫，共同霍普夫，单端，字双曲群。
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…
We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for $\underline{E}G$ and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one relator groups, the Hilbert modular group and $3$-manifold groups.
Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions
本文应用拓扑方法证明特殊幺半群从其单位群继承有限性性质，并据此证明单关系幺半群的 Lyndon 恒等式定理，同时发展 Bass-Serre 理论研究融合自由积与 HNN 扩张的闭包性质。
Short presentations for crystallographic groups
本文提出一种构造欧几里得晶体学群简短表示的实用方法，强调关系子与凯莱图循环的联系，通过实例验证其与强环的对应关系，并计算了多维度周期图上顶点传递群的表示。
Small Cancellation Stability and Isomorphism Rigidity for Generic Finitely Presented Groups
该研究证明一般有限表现群的同构刚性，给出一般情形下求解m生成q关系群同构问题的二次时间算法，并确定对应同构类型数量的渐近表达式。
We investigate conformal dimension for the class of infinite hyperbolic groups in the Gromov density model $\mathcal{G}^d_{m,l}$ of random groups with $m \geq 2$ fixed generators, density $0 < d < 1/2$ and relator length $l \to \infty$. Our main result is a lower bound linear in $l$ at all densities $0 < d < 1/2$ achieved by building undistorted round trees coming directly from lower density Gromov random groups.
Group presentations are implicit descriptions of 2-dimensional cell complexes with only one vertex. While such complexes are usually sufficient for topological investigations of groups, multi-vertex complexes are often preferable when the focus shifts to geometric considerations. In this article, I show how to quickly describe the most important multi-vertex 2-complexes using a slight variation of the traditional group presentation. As an illustration I describe multi-vertex 2-complexes for torus knot groups and one-relator Artin groups from which their elementary properties are easily derived…
For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any density less than a half (or in the few relators model) contains quasiconvex subgroups commensurable with some member of the family, with overwhelming probability.
Fundamental groups and group presentations with bounded relator lengths
本文研究具有平凡第一贝蒂数的紧致测地空间在有限群等距作用下的直径比上界，并针对有限阿贝尔群的凯莱图，利用二维 CW 复形 $P_k(Γ(G,S))$ 给出拉普拉斯第一非平凡特征值、直径、切格常数和卡日丹常数的显式界。
Cone types and asymptotic invariants for the random walk on the modular group
本文通过刻画模群凯莱图的锥类型并计算增长函数，进而估计随机游走的谱半径、熵和漂移等渐近不变量。
Growth of the asymptotic dimension function for groups
构造一个渐近维数为有限生成群是相对容易的：$\mathbb{Z}$与$\mathbb{Z}$的限制圈积就是一个例子。有鉴于此，考虑群的渐近维数函数的增长率变得有趣。不严格地说，我们在$λ$尺度上测量维数，并让$λ$增加到无穷大以恢复渐近维数。在本文中，我们考虑如何渐近维数函数的影响，不同的建设涉及的群体。
Asymptotic dimension of relatively hyperbolic groups
假设一个有限生成群$G$相对于一个子群集合$\{H_1, ..., H_m\} $是双曲的。证明了如果每个子群$H_1, ..., H_m$都有有限的渐近维数，则$G$的渐近维数也是有限的.
Asymptotic properties of groups acting on complexes
我们研究渐近维数和性质A的群体作用于复杂的。特别地，我们证明了一个有限的，可发展的复杂的群体的基本群将有有限的渐近维数提供的几何实现的发展有有限的渐近维数和顶点群是双生成的，并有有限的渐近维数。我们还证明了性质A是保持这种结构的几何实现的发展有有限的渐近维数和顶点群都有性质A。这些结果自然地推广了群的图的基本群保持这些大尺度性质的相应结果。我们还用一个例子来说明发展有有限渐近维数的要求不能放松。
On asymptotic dimension of groups acting on trees
本文证明了如下定理：设$π$是有限群图的基本群，该有限群图的顶点群$G_v$对所有顶点$v$都具有asdim $G_v\le n$.然后是asdim $π\le n+1$。这给出了HNN扩展和合并产品的渐近维数的最佳估计。
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…
Semigroup-controlled asymptotic dimension
引入了半群控制渐近维数的概念。这个概念推广了大尺度下的渐近维数和渐近Assouad-Nagata维数。在小尺度和全局尺度下也存在半群控制维数。半群控制的渐近维数满足渐近维数论的许多基本性质。我们研究这些新的维度如何有助于理解粗嵌入和均匀嵌入。特别地，我们引入了不可数许多准等距下的不变量和不可数许多双Lipschitz不变量。Hurewicz型定理的推广和几何群论的一些应用。
We prove that a metric space with subexponential asymptotic dimension growth has Yu's property A.
The Asymptotic Dimension of the First Grigorchuk Group is Infinity
我们证明了第一Grigorchuk群的渐近维数是无穷大。
On asymptotic dimension of countable abelian groups
在不变的真度量下，我们计算了有理数的渐近维数。另外，我们还证明了一个可数挠阿贝尔群具有不变的真度量时，它的渐近维数为零。
A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory
证明了关于低维映射的经典Hurewicz定理的一个渐近类似定理.这个定理使我们能够找到尖锐的上界估计的渐近维数的群作用在有限维度量空间，并允许我们证明一个有用的推广定理的渐近维数。作为应用，我们发现的上界估计的渐近维数的幂零和多环组的赫希长度。我们也能够提高已知的上界的渐近维数的基本群的复群，合并的自由积和双曲化的度量空间具有Higson性质。
We give a characterization for asymptotic dimension growth. We apply it to CAT(0) cube complexes of finite dimension, giving an alternative proof of N. Wright's result on their finite asymptotic dimension. We also apply our new characterization to geodesic coarse median spaces of finite rank and establish that they have subexponential asymptotic dimension growth. This strengthens a recent result of J. Spakula and N. Wright.
We show that every box space of a virtually nilpotent group has asymptotic dimension equal to the Hirsch length of that group.
In this paper we prove that the asymptotic dimension of a finite-dimensional CAT(0) cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every CAT(0) cube complex is a contractive retraction of an infinite dimensional cube. As an example of the dimension theorem we obtain bounds on the asymptotic dimension of small cancellation groups.
Asymptotic dimension and uniform embeddings
我们证明了有限渐近维数空间的型函数估计其Hilbert（或任何$l^p$）压缩。该方法允许获得压缩的lampighter组$Z\wr Z$，它具有无穷渐近维数的下界。
The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "$N$-$\mathcal F$-amenability" or "amenability dimension", and "$d$-BLR condition") and its generalisation, transfer reducibility, which are versions of asymptotic dimension invented for the proofs of the Farrell--Jones and Borel conjectures. We prove that groups of null equivariant asymptotic dimension are exactly virtually cyclic groups. We show that a covering of the boundary always extends to a covering of the whole compactification. We provide a number of characterisations of equivariant asymptot…
On asymptotic dimension of groups
证明了渐近维数的可数并定理的一个版本，并将其应用于作用于渐近有限维度量空间上的群。作为结果，我们得到以下有限维定理。A）渐近有限维群的合并积具有有限的渐近维数：asdim A *_C B < infinity。B）设G'是群G的HNN扩张，且asdim G < infinity。则asdim G '<infinity。C）设Γ是由群π构造的Davis群，其中asdimπ<无穷大。则asdim r <无穷大。
Asymptotic Dimension of Discrete Groups
我们将Gromov关于有限生成群的渐近维数的概念推广到所有离散群。特别地，我们将[B-D2]中证明的Hurewicz型定理推广到一般群。然后我们利用这个推广证明了一个关于群生成的可解群的渐近维数的公式。
Asymptotic dimension of proper CAT(0) spaces which are homeomorphic to the plane
本文研究了一个与$R^2$同胚的真CAT（0）空间$(X,d)$，并证明了它的渐近维数$asdim (X,d)$等于2。
Mapping class groups have finite asymptotic dimension
通过把它们看作群的可展复形的基本群，我们证明了紧致可定向曲面的映射类群具有有限的渐近维数。
Asymptotic Dimension
渐近维数论是由Gromov在90年代初创立的。在本文中，我们给一个调查的最近的历史，我们强调它的两个特点：类比与维数论紧凑的度量空间和应用理论的离散群。
On asymptotic dimension of amalgamated products and right-angled Coxeter groups
证明了不等式$$ \as A\ast_CB\le\max\{\as A,\as B,\as C+1\} $$，并应用这个不等式证明了任意直角Coxeter群的渐近维数不超过其Davis复形的维数.
Dimension of asymptotic cones of Lie groups
计算了连通李群的渐近锥面的覆盖维数.对于单连通可解李群，这是指数根的余维数。作为证明的一个应用，我们给出了拟等距嵌入非正曲度量空间的连通李群的一个刻画。
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。
We show that the asymptotic dimension of box spaces behaves (sub)additively with respect to extensions of groups. As a result, we obtain that for an elementary amenable group, the asymptotic dimension of any of its box spaces is bounded above by its Hirsch length. This bound is shown to be an equality for a large subclass of groups including all virtually polycyclic groups.
A note on spaces of asymptotic dimension one
设$X$是一个测地度量空间，其$H_1(X)$是一致生成的.如果$X$的渐近维数为1，则$X$是无界树的拟等距。作为推论，我们证明了具有亏格$g \ge 2$和一个边界分量的紧致定向曲面的曲线图的渐近维数至少为2。
Asymptotic dimension and geometric decompositions in dimensions 3 and 4
本文证明Thurston几何分解的4维流形基本群渐近维数至多4，闭3维流形群至多3，并由此推出Novikov猜想成立及若干几何推论。
Asymptotic properties of finitely generated subgroups of free groups, and of finite group presentations, can be considered in several fashions, depending on the way these objects are represented and on the distribution assumed on these representations: here we assume that they are represented by tuples of reduced words (generators of a subgroup) or of cyclically reduced words (relators). Classical models consider fixed size tuples of words (e.g. the few-generator model) or exponential size tuples (e.g. Gromov's density model), and they usually consider that equal length words are equally likel…
We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of controlled topology, and its connections with Gromov's theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for C*-algebras associated to dynamical systems. Dynamic asymptotic dimension also …
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