A. Yu. Olshanskii, M. V. Sapir
We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann's problem. Our group is an extension of a group of finite exponent n >> 1 by a cyclic group, so it satisfies the identity [x,y]^n = 1.
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The goal of this article is to study results and examples concerning finitely presented covers of finitely generated amenable groups. We collect examples of groups $G$ with the following properties: (i) $G$ is finitely generated, (ii) $G$ is amenable, e.g. of intermediate growth, (iii) any finitely presented group $E$ with a quotient isomorphic to $G$ contains non-abelian free subgroups, or the stronger (iii') any finitely presented group with a quotient isomorphic to $G$ is large.
We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among $G$-CW-complexes for these groups. As another application we prove that the $L^2$-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.
In this note, we construct torsion-free countable, amenable, weakly mixing groups, which answer a question of V. Bergelson. Some results related to verbal subgroups and crystallographic groups are also presented.
We prove that the approximation conjecture of Luck holds for all amenable groups in the complex group algebra case. This result was previously proved by Dodziuk, Linnell, Mathai, Schick and Yates under the assumption that the group is torsion-free elementary amenable.
We introduce and investigate different definitions of effective amenability, in terms of computability of Følner sets, Reiter functions, and Følner functions. As a consequence, we prove that recursively presented amenable groups have subrecursive Følner function, answering a question of Gromov, for the same class of groups we prove that solvability of the Equality Problem on a generic set (generic EP) is equivalent to solvability of the Word Problem on the whole group (WP), thus providing the first examples of finitely presented groups with unsolvable generic EP. In particular, we prove that f…
We construct an example of a finitely-generated amenable group that does not admit any coarse 1-Lipschitz embedding with positive compression exponent into L_p for any 1 \leq p < \infty, answering positively a question of Arzhantseva, Guba and Sapir.
Torsion homology growth and cheap rebuilding of inner-amenable groups
本研究利用Tucker-Drob的内顺从群结构定理，证明了实质上无挠、剩余有限的非顺从内顺从群具有廉价1-重建性质，推导其第一ℓ²-Betti数与1次对数挠为零，将相关结果从顺从群推广到内顺从群。
We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-$p$ groups can grow arbitrarily fast. By way of contrast in $\mathbb Z_p$- analytic groups the torsion growth is at most polynomial.
We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.
Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $\tilde X$ with compact quotient $X$. Fix $n \geq 1$, assume $H_n(\tilde X, \mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We prove that the torsion of the integral homology in dimension $n$ of $\tilde{X}/H_i$ grows subexponentially in $[G:H_i]$. By way of contrast, if $X$ is not compact, there are solvable groups of derived length 3 for which torsion in homology can grow faster than any given function.
Finitely generated infinite torsion groups that are residually finite simple
本文证明每个有限生成剩余有限挠群可嵌入某个有限生成且剩余有限单的挠群，从而构造出有限生成无限挠的剩余有限单群，回答了 Olshanskii 与 Osin 的问题。
We prove that any finitely generated elementary amenable group of zero (algebraic) entropy contains a nilpotent subgroup of finite index or, equivalently, any finitely generated elementary amenable group of exponential growth is of uniformly exponential growth. We also show that 0 is an accumulation point of the set of entropies of elementary amenable groups.
We introduce right generating sets, Cayley graphs, growth functions, types and rates, and isoperimetric constants for left homogeneous spaces equipped with coordinate systems; characterise right amenable finitely right generated left homogeneous spaces with finite stabilisers as those whose isoperimetric constant is $0$; and prove that finitely right generated left homogeneous spaces with finite stabilisers of sub-exponential growth are right amenable, in particular, quotient sets of groups of sub-exponential growth by finite subgroups are right amenable.
A famous result of Hall asserts that the multiplication and exponentiation in finitely generated torsion free nilpotent groups can be described by rational polynomials. We describe an algorithm to determine such polynomials for all torsion free nilpotent groups of given Hirsch length. We apply this to determine the Hall polynomials for all such groups of Hirsch length at most 7.
In this paper we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups $G$ into unitriangular groups $UT_n(\mathbb{Z})$. We also provide a polynomial time algorithm for finding distortion of a given subgroup of $G$
We show that a construction by Aanderaa and Cohen used in their proof of the Higman Embedding Theorem preserves torsion length. We give a new construction showing that every finitely presented group is the quotient of some $C'(1/6)$ finitely presented group by the subgroup generated by its torsion elements. We use these results to show there is a finitely presented group with infinite torsion length which is $C'(1/6)$, and thus word-hyperbolic and virtually torsion-free.
In his work on the Farrell-Jones Conjecture, Arthur Bartels introduced the concept of a "finitely $\mathcal{F}$-amenable" group action, where $\mathcal{F}$ is a family of subgroups. We show how a finitely $\mathcal{F}$-amenable action of a countable group $G$ on a compact metric space, where the asymptotic dimensions of the elements of $\mathcal{F}$ are bounded from above, gives an upper bound for the asymptotic dimension of $G$ viewed as a metric space with a proper left invariant metric. We generalize this to families $\mathcal{F}$ whose elements are contained in a collection, $\mathfrak{C}$…
A finitely generated ICC non property (T) group that is neither inner amenable nor properly proximal
本文构造了Caprace非内可均群的有限生成扩张，该扩张为GL_4(F_2(t))的ICC子群，无性质(T)，且非适当近端，从而解决了Boutonnet等人提出的问题。
$R_{\infty}$-property for finitely generated torsion-free 2-step nilpotent groups of small Hirsch length
本文研究了有限生成无挠2步幂零群的$R_{\infty}$-性质，证明Hirsch长度至多为6的群不具有该性质，而长度为7的群存在具有该性质的例子。
We give a uniform construction that, on input of a recursive presentation $P$ of a group, outputs a recursive presentation of a torsion-free group, isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain a known result, the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed. We apply our techniques to show that recognising embeddability of finitely presented groups is $Π^{0}_{2}$-hard, $Σ^{0}_{2}$-hard, and lies in $Σ^{0}_{3}$. We also show that the sets of orders of torsion elements of finitel…
We construct an embedding of a free Burnside group $B(m,n)$ of odd $n > 2^{48}$ and rank $m >1$ in a finitely presented group with some special properties. The main application of this embedding is an easy construction of finitely presented non-amenable groups without noncyclic free subgroups (which provides a finitely presented counterexample to the von Neumann problem on amenable groups). As another application, we construct weakly finitely presented groups of odd exponent $n \gg 1$ which are not locally finite.
We construct the first examples of finitely presented groups with cubic Dehn function containing a finitely generated infinite torsion subgroup. Moreover, we show that any infinite free Burnside group with sufficiently large odd exponent can be embedded as a subgroup of a finitely presented group with cubic Dehn function.
We prove a new weak mean ergodic theorem (Theorem A) for 1-cocycles associated to weakly mixing representations of amenable groups. Let $G$ be a finitely generated, discrete, amenable group $G$ which admits a controlled Folner sequence. We use Theorem A to deduce that any affine action $G\ca^T \Cal H$ on Hilbert space with weakly mixing linear part admits a sequence of almost fixed points (Theorem B). Specializing to the case that $G$ is a finitely generated group of polynomial growth, we show that convex combinations of averages of the associated 1-cocycle over $n$-balls provide a sequence of…
We show that every finitely-generated non-amenable linear group over a field of characteristic zero admits an ergodic action which is rigid in the sense of Popa. If this group has trivial solvable radical, we prove that these actions can be chosen to be free. Moreover, we give a positive answer to a question raised by Ioana and Shalom concerning the existence of such a free action for $\mathbb{F}_2\times\Z$. More generally, we show that for groups $Γ$ considered by Fernos in \cite{Fer}, e.g. Zariski-dense subgroups in PSL$_n(\R)$, the product group $Γ\times \Z$ admit a free ergodic rigid actio…
On self-similarity of finitely generated torsion-free nilpotent groups
本文研究有限生成无挠幂零群的自相似性，建立了几何等价性与自相似性的联系，证明所有2生成3类无挠幂零群可忠实表示为传递自相似群，并给出一个4生成3类反例，解决了Sidki提出的问题。
Uniformly finite homology is a coarse invariant for metric spaces; in particular, it is a quasi-isometry invariant for finitely generated groups. In this article, we study uniformly finite homology of finitely generated amenable groups and prove that it is infinite dimensional in many cases. The main idea is to use different transfer maps to distinguish between classes in uniformly finite homology. Furthermore we show that there are infinitely many classes in degree zero that cannot be detected by means.
We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-amenable boundedly acyclic groups and construct a finitely presented non-amenable boundedly acyclic group. On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has uncountable dimension in all degrees greater than or equal to~$2$, and a concrete finitely presented one. Countable non-amenable groups with these two extre…
We construct a finitely presented torsion-free simple group $Σ_0$, acting cocompactly on a product of two regular trees. An infinite family of such groups has been introduced by Burger-Mozes ([2,4]). We refine their methods and get $Σ_0$ as an index 4 subgroup of a group $Σ< \mathrm{Aut}(\mathcal{T}_{12}) \times \mathrm{Aut}(\mathcal{T}_{8})$ presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [4, Theorem 6.4] needs more than 18000 relations, and the smallest simple group constructed in [4, Section 6.5] needs even more than 360000 relations…
We give a complete survey of a construction by Boone and Collins for embedding any finitely presented group into one with $8$ generators and $26$ relations. We show that this embedding preserves the set of orders of torsion elements, and in particular torsion-freeness. We combine this with the independent results of Belegradek and Chiodo to prove that there is an $8$-generator $26$-relator universal finitely presented torsion-free group (one into which all finitely presented torsion-free groups embed).
We study the growth of torsion in the abelianizations of finite index subgroups in finitely generated metabelian groups. This complements earlier work of Kar, Kropholler and the author which covered the finitely presented amenable groups.
From telescopes to frames and simple groups
本文引入群的望远镜概念，模仿分支群构造得到若干 B-望远镜例子，并据此构造出 2-生成无限 amenable 单群、LEF 单群嵌入、有限单群直积中的 2-生成框架及 Grothendieck 对等应用。
We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are "optimal" in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any $n\geq2^{48}$ such that $n$ is either odd or divisible by $2^9$, any infinite free Burnside group with exponent $n$ is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.
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。
We generalize a result of R. Thomas to establish the non-vanishing of the first l2-Betti number for a class of finitely generated groups.
A simple construction of infinite finitely generated torsion groups
本文利用 Nielsen-Schreier 指数公式，为 Golod 关于存在无限有限生成挠群的定理提供了一个足够简单、可纳入标准群论课程的新证明。
A note on torsion length and torsion subgroups
本文通过构造具有任意挠长的polycyclic群及挠子群商群非有限展示的有限展示群示例，解答了Kourovka笔记本中的两个问题。
Some elementary amenable subgroups of interval exchange transformations
本文研究了区间交换变换中某些初等可容许子群的性质，探讨了其非虚拟可解性和可解性条件，并展示了无限多个非线性且非虚拟可解的群以及可解群的导出长度。
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.
This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenabil…
It is well known that any polycyclic group, and hence any finitely generated nilpotent group, can be embedded into $GL_{n}(\mathbb{Z})$ for an appropriate $n\in \mathbb{N}$; that is, each element in the group has a unique matrix representation. An algorithm to determine this embedding was presented in Nickel. In this paper, we determine the complexity of the crux of the algorithm and the dimension of the matrices produced as well as provide a modification of the algorithm presented by Nickel.
Strongly 1-bounded inner amenable groups
本文研究了有限生成icc内可 amenable群，证明其生成强1有界的II_1因子。
We consider a finitely generated group acting minimally on a compact space by homeomorphsims, and assume that the Schreier graph of at least one orbit is quasi-isometric to a line. We show that the topological full group of such an action is amenable.
We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $ε_0 +1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In fact we also obtain, for each $α< ε_0$, a finitely generated elementary amenable subgroup of $F$ whose EA-class is $α+ 2$. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with $\mathbf{Z} + \mathbf{Z}$, $\mathbf{Z} \wr \mathbf{Z}$, a…
It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup is abelian.
Chung and Jiang showed that, if a one ended group contains an infinite order element, then every continuous cocycle over the full shift on that group, taking values in a discrete group, must be cohomologous to a homomorphism. We show that their conclusion holds for all one ended groups, so that the hypothesis of admitting an infinite order element may be omitted.
We consider groups of automorphisms of locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. This covers all known examples of groups that are not elementary amenable and act on the trees: groups of intermediate growths and Basilica group, by giving a more straightforward proof. Moreover, we deduce that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.
We construct a finitely generated solvable subgroup of Homeo(R) with a non-metaabelian characterizing quotient.
Residually finite amenable groups that are not Hilbert-Schmidt stable
该研究构造了首批不具备Hilbert-Schmidt稳定性的剩余有限顺从群实例，涵盖多种群结构，还证明这些实例不具备算子HS稳定性，填补了相关领域的研究空白。
In this paper, a group is called weakly amenable if its left regular representation is not uniformly isolated from the trivial representation. First examples of finitely generated non-amenable weakly amenable groups are constructed.
We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.
Realizing residually finite groups as subgroups of branch groups
本文证明任意有限生成剩余有限群均可嵌入保持挠性、可顺从性等性质的有限生成完全分支群，并据此构造有限生成不可顺从挠分支群。
We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich group, a finitely presented solvable group with unsolvable word problem, has computable Følner sets. We also prove computability of Følner sets for a group that is extension of an amenable group with solvable word problem by a finitely generated group with computable Følner sets with subrecursive distortion function. Moreover we obtain some known and some new upper bounds for the Følner function in these particular extensions.
We show that for any amenable group Γand any ZΓ-module M of type FL with vanishing Euler characteristic, the entropy of the natural Γ-action on the Pontryagin dual of M is equal to the L2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module ZΓ/ZΓf is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in ZΓ. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szegő-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra …
We provide the first examples of finitely generated simple groups that are amenable (and infinite). This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by constructing a suitable family of densities on the classical Bernoulli space.
We show that finitely generated Abelian groups admit equal growth functions with respect to symmetric generating sets if and only if they have the same rank and the torsion parts have the same parity. In contrast, finitely generated Abelian groups admit equal growth functions with respect to monoid generating sets if and only if they have same rank. Moreover, we show that the size of the torsion part is in fact determined by the set of all growth functions of a finitely generated Abelian group.
Using probabilistic methods, Collins and Dykema proved that the free product of two sofic groups amalgamated over a monotileably amenable subgroup is sofic as well. We show that the restriction is unnecessary; the free product of two sofic groups amalgamated over an arbitrary amenable subgroup is sofic. We also prove a group theoretical analogue of a result of Kenley Jung. A finitely generated group is amenable if and only if it has only one sofic representation up to conjugacy equivalence.
Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $Γ$, does there exist a generating set $S$ such that the Cayley graph $(Γ,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(Γ,S)<p_u(Γ,S)$? We show that this is true if $Γ$ contains an infinite normal subgroup $N$ such that $Γ/ N$ is non-amenable. Moreover for any finitely generated group $G$ containing $Γ$ there exists a generating set $S'$ of $G$ such that $p_c(G,S')<p_u(G,S')$. In particular this applies to fr…
Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Gamma has a faithful action on the circle, then some finite-index subgroup of Gamma has the property that all of its nontrivial, finitely generated subgroups have infinite, cyclic quotients. It also means that every left-orderable, amenable group is locally indicable. This answers a question of P Linnell.
The product formula for Reidemeister numbers on nilpotent groups
该研究针对有限生成无挠幂零群上的Reidemeister数乘积公式，将其推广至中心扩张，推导相关一般性结论并构造出一族具有满Reidemeister谱的无穷多群。
We show that any one-relator group $G=F/\langle\langle w\rangle\rangle$ with torsion is coherent -- i.e., that every finitely generated subgroup of $G$ is finitely presented -- answering a 1974 question of Baumslag in this case.
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