Rank gradient and torsion groups
D. Osin
构造了具有正秩梯度的无穷余生成剩余有限扭群的第一个例子。特别是，这些群体是不服从的。讨论了它在成本和$L^2$ -贝蒂数问题中的应用.
We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient. In particular, these groups are non-amenable. Some applications to problems about cost and $L^2$-Betti numbers are discussed.
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On controllers of prime ideals in group algebras of torsion-free abelian groups of finite rank
本文考虑了有限秩阿贝尔群的群代数中素理想的一些研究方法
Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.
We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$ subgroups of finite index in $G$.
It is known that every torsion-free abelian group of finite rank has a maximal completely decomposable summand that is unique up to isomorphism. We show that groups of infinite rank need not have maximal completely decomposable summands, but when they do, this summand is unique up to isomorphism.
Let p be a prime and G be a torsion-free abelian group. A homomorphism from G to the p-adic integers is called a p-adic functional on G. If G has finite rank, then G can be represented as an inductive limit of an inductive sequence of free abelian groups of the same rank, and the group of all p-adic functionals on G is described in terms of this inductive sequence. If this inductive sequence is stationary--i.e., if the homomorphism is the same at every stage--then the group of p-adic functionals is described in particularly simple terms, as a right-submodule that is invariant under the module …
Semisimple torsion in groups of finite Morley rank
本文证明了关于有限Morley秩群的几个结果：p-挠总是发生在环面内，Sylow p-子群是共轭的，p不是我们逼近到“外尔群的最小素除子.这些结果很快就在分类项目中得到了广泛的应用。
Bounded rank subgroups of Coxeter groups, Artin groups and one-relator groups with torsion
我们得到了Coxeter群、Artin群和带挠的单关系群的自由性、拟凸性和可分性的一些结果。
We obtain sufficient criteria for endomorphisms of torsion-free nilpotent groups of finite rank to be automorphisms, by considering the induced maps on the torsion-free abelianisation and the centre. Whilst these results are known in the finitely generated case removing this assumption introduces several difficulties.
Some torsion-free solvable groups with few subquotients
本研究构造出满足所有有限生成无挠亚阿贝尔子商均为虚阿贝尔的无限秩有限生成无挠可解群，证明了P. Kropholler的无限秩可解群刻画定理不存在无挠版本。
An indecomposable decomposition of a torsion-free abelian group $G$ of rank $n$ is a decomposition $G=A_1\oplus\cdots\oplus A_t$ where $A_i$ is indecomposable of rank $r_i$ so that $\sum_i r_i=n$ is a partition of $n$. The group $G$ may have decompositions that result in different partitions of $n$. We address the problem of characterising those sets of partitions of $n$ which can arise from indecomposable decompositions of a torsion-free abelian group.
The cohomology of virtually torsion-free solvable groups of finite rank
设$G$是一个有限秩的虚无挠的可解群，$A$是一个其基础阿贝尔群是无挠的且有有限秩的$\mathbb ZG$ -模.我们在$A$上规定了一个条件，确保$H^n(G,A)$和$H_n(G,A)$对于所有$n\geq 0$都是有限的。利用二维上同调的这一性质，我们推出了关于有限秩可解群中存在近补和补的两个结果。作为近补定理的应用，我们得到了关于可解群的同调维数的一个新结果。
For an Abelian group $G$, any homomorphism $μ\colon G\otimes G\rightarrow G$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}\,G$ of all multiplications on an Abelian group $G$ itself is an Abelian group with respect to addition; the group is called the \textsf{multiplication group} of $G$. Let $\mathcal{A}_0$ be the class of all reduced block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, for groups $G\in \mathcal{A}_0$, we describe groups $\text{Mult}\,G$. We prove that for $G\in \mathcal{A}_0$, the group $\text{Mult}\,…
The ultrasimplicial property for simple dimension groups with unique state, the image of which has rank one
令$G$是一个序群，它是秩1无挠阿贝尔群和有限秩无挠阿贝尔群的直和，其序结构由第一被加数上的自然序产生。给出了$G$具有序群归纳极限表示的一个必要条件和一个充分条件.
Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G contains a direct sum of strongly indecomposable groups as a characteristic subgroup of finite index, giving rise to a classification of finite rank strongly indecomposable torsion--free abelian groups.
Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the trivial element, then $g$ is called a generalized torsion element. To the best of our knowledge, we have no hyperbolic $3$--manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer $n > 1$ there are infinitely many closed hyperbolic $3$--manifolds $M_n$ which enjoy the property: (i) the Heegaard genus of $M_n$ is $n$, (ii) the rank of the fundamental grou…
Torsion cohomology for solvable groups of finite rank
我们定义了一类有限阿贝尔节秩的可解群$\mathcal{U}$，它包括所有这样的群，几乎是无挠的，以及那些是无挠生成的。假设$G$是$\mathcal{U}$中的一个组，$A$是$\mathbb ZG$ -模块。如果$A$是$\mathbb Z$ -无挠的，并且具有有限的$\mathbb Z$ -秩，则我们在$A$上规定一个条件，该条件保证对于$n\geq 0$，$H^n(G,A)$和$H_n(G,A)$必须是有限的。此外，如果$A$的基础阿贝尔群是一个Černikov群，我们确定了一个类似的条件，确保$H^n(G,A)$必须是一个Černikov群的所有$n\geq 0$。
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most non-uniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (co-co…
Asymptotic behaviour of analytic torsion and cohomological torsion for $\mathbb{Q}$-rank $1$ arithmetic groups
本文扩展了先前工作中关于 SL(n) 同余子群的解析 torsion 的精细渐进行为，应用于获得 SL(2, O_F) 同余子群和 SO(n,1) 同余子群的上同调 torsion 增长的新渐进行为和界。
The problem of the classification of the nilpotent class 2 torsion free groups up to the geometrically equivalence
本文研究了2类幂零生成的无挠无挠的群在几何等价下的分类问题。用一种很简单的方法证明了这个问题等价于完全（在Maltsev意义下）幂零无无挠的有限秩群的分类问题直到同构.这个结果，使我们能够再一次理解幂零类2群的拟簇的分类问题的复杂性。众所周知，幂零类s（对于每一个s）群的簇是诺特群。因此，即使由单个幂零类2生成的无挠无挠的群生成的拟簇的分类问题，等价于直到同构的完全（在Maltsev意义下）无挠无挠的有限秩群的分类问题。
Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut(G) on the set of maximal independent subsets of G determine the indecomposable decompositions of G. G contains a direct sum of pure strongly indecomposable groups as a subgroup of finite index.
In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $\ell$-subgroups in any finite group of Lie type, for any prime $\ell$, which may be of independent interest.
On the p-rank of Ext(A,B) for countable abelian groups A and B
本文证明了可数无挠阿贝尔群A和B的Ext（A，B）的p秩是可数的或连续统的大小.
A coarse classification of countable abelian groups
我们对有限无挠的秩的所有可数阿贝尔群进行分类直到粗等价。Q-上同调维数和无挠的秩是给出这种分类的两个不变量。证明了任意有限无挠的秩的可数阿贝尔群粗略等价于Z^n + H，其中H是循环群的直和（可能是无限的）.本文将阿贝尔群的Gromov刚性定理部分推广到可数阿贝尔群。
We prove that a self-similar free abelian group has finite rank. We apply the result to self-similar wreath products of abelian groups $G=BwrX$. We show that if $X$ is torsion-free, then $B$ is torsion of finite exponent. Furthemore, we construct a self-similar group $G=BwrC_{2}$ where $B$ is free abelian of infinite rank.
Groups of finite Morley rank with solvable local subgroups
我们奠定了基础的理论群体的有限莫雷秩在当地的子群是可解的，我们进行本地分析这些群体。我们证明了类似于有限群理论中Bender方法的主唯一性定理，并导出了它的推论。我们也考虑齐次情形以及扭转。
Arbitrary p-Gradient Values
对任意素数p和任意正实数α，构造了一个p-梯度等于α的有限生成群Γ.这个构造用来证明存在不可数多个成对不可分解的群，它们是可分解生成的、无限的、挠的、不可修正的和剩余p的。
Rank gradient and p-gradient of amalgamated free products and HNN extensions
我们计算了自由积、合并后的自由积以及合并后的HNN扩张的秩梯度和p-梯度。成本的概念被用来计算混合自由积和HNN扩展的秩梯度。对于p-梯度，将使用合并自由积和HNN扩展的Kurosh子群定理。
Which finitely generated Abelian groups admit equal growth functions?
证明了对称生成的阿贝尔群关于对称生成集的增长函数相等当且仅当它们具有相同的秩且扭部分具有相同的奇偶性。与此相反，幺半群生成的阿贝尔群允许关于幺半群生成集的等增长函数当且仅当它们具有相同的秩。此外，我们还证明了挠率部分的大小实际上是由一个生成的阿贝尔群的所有增长函数的集合决定的。
Let $G$ be a finitely generated solvable-by-finite linear group. We present an algorithm to compute the torsion-free rank of $G$ and a bound on the Prüfer rank of $G$. This yields in turn an algorithm to decide whether a finitely generated subgroup of $G$ has finite index. The algorithms are implemented in MAGMA for groups over algebraic number fields.
We prove that a profinite group $G$ with positive rank gradient does not satisfy a group law. In the case when $G$ is a pro-$p$ group we show that $G$ contains a nonabelian dense free subgroup.
Sylow 0-unipotent subgroups in groups of finite Morley rank
简单代数群分类的一个主要工具是区分半单群和幂幺群。对于有限Morley秩群的无挠子群，如何进行这种区分并不是先验清楚的。利用0-幺正性的“分次”概念，我们发展了有限Morley秩的可解群的无挠子群的Sylow理论。这Sylow理论提供了一个强大的替代通常的理论卡特子群，并将用于分析的交集的博雷尔子群在极小单群。
We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.
Generalized Bassian and other Mixed Abelian Groups with Bounded p-Torsion
本文刻画了具有有界p-挠的混合阿贝尔群，证明所有广义Bassian群具有有限挠自由秩，从而解答了Chekhlov-Danchev-Goldsmith的公开问题，并揭示了其与B+E群的等价及子群关系。
On groups and fields interpretable in torsion-free hyperbolic groups
我们证明了非循环无挠的双曲群G的一般类型对于任何可解释的阿贝尔群都是外来的，因此对于任何可解释的域也是外来的。这一结果取决于，除其他事项外，在可定义的简单性的非循环无挠的双曲群，我们借此机会给一个证明，后者使用Sela的描述mandaries在无挠的双曲群。我们还利用代数的描述证明了：如果F是秩> 2的自由群，则F的有限元组在Aut（F）下的轨道是不可定义的.
Sparse Random Covers and Growth of Torsion in First Homology
该研究通过支架泊松过程构造高阶局部对称空间的随机开覆盖，证明高阶对称空间对应无挠格序列的一阶同调归一化挠率对数与体积比值趋于0，回答了相关问题并证实了对应猜想，还得到了定量界及仿射建筑的类似结论。
A characterization of Ext(G,Z) assuming V=L
本文在Godel可构造性公理下，对任意挠自由的阿贝尔群G，完成了Ext（G，Z）的刻画.特别地，我们证明了在（V=L）中，对于小于第一弱紧基数的不可数共尾性的奇异基数nu，且对于每个基数序列（nu_p：p在P中）满足nu_p <= 2^{nu}，则存在一个无挠的阿贝尔群G，其大小为nu，使得nu_p等于Ext（G，Z），2^{nu}是Ext（G，Z）的无挠秩。
Let $P_k$ be the subgroup generated by $k$th powers of primitive elements in $F_r$, the free group of rank $r$. We show that $F_2/P_k$ is finite if and only if $k$ is $1$, $2$, or $3$. We also fully characterize $F_2/P_k$ for $k = 2,3,4$. In particular, we give a faithful nine dimensional representation of $F_2/P_4$ with infinite image.
Kleinian groups and the rank problem
证明了无挠字双曲Kleinian群类的秩问题是可判定的。我们还表明，在这个类中的每一个组只有1000个尼尔森等价类的生成集的一个给定的基数。
We observe that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups.
Rank gradient, cost of groups and the rank versus Heegaard genus problem
本文研究了剩余有限群中指数有限的子群的秩的增长，并将其与代价的概念联系起来。作为一个副产品，我们证明了双曲3-流形上的“秩与Heegaard属”猜想与拓扑动力学中的“固定价格问题”是不相容的。
Geometrical equivalence of nilpotent torsion free groups
幂零群的簇是Noether簇。这就是为什么两个幂零类群是几何等价的当且仅当它们有相同的拟恒等式（[Pl3]）。因此，我们可以描述幂零群群的几何等价类，而不是描述单个幂零群生成的拟簇。几何等价问题可以用群的逼近技巧来研究（[Pl2]）。本文只考虑幂零的无挠无挠的群。第一部分给出了幂零无挠的群与其Mal'tsev完备化几何等价从而属于同一拟簇的两个充分条件（定理1和定理2）。本文的第二部分利用群的逼近的简便方法，得到了中心秩较小的幂零2类无挠的群的几何等价类的刻划。
Normal generation and $\ell^2$-betti numbers of groups
群的\emph{normal rank}是其正规闭包与群重合的元素的最小数目。研究了群的正规秩与其第一个$\ell^2$ - Betti数之间的关系，并猜想对于无挠群，不等式$β_1^{(2)}(G)$不超过正规秩负1.证明了左可序顺从群极限的猜想。另一方面，对于每个$n\ge 2$和每个$\e>0$，我们给出一个简单群$Q$（具有挠）的例子，使得$β_1^{(2)}(Q) \geq n-1-ε$。这些群也提供了每个$n\ge 2$的秩恰好为$n$的简单群的例子;直到现在还不知道$n> 3$存在这样的例子。
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