Algebraic entropy of elementary amenable groups

Elementary amenable groups are quasidiagonal

We show that the group C*-algebra of any elementary amenable group is quasidiagonal. This is an offspring of recent progress in the classification theory of nuclear C*-algebras.

On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups

We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy.

Chain conditions, elementary amenable groups, and descriptive set theory

We first consider three well-known chain conditions in the space of marked groups: the minimal condition on centralizers, the maximal condition on subgroups, and the maximal condition on normal subgroups. For each condition, we produce a characterization in terms of well-founded descriptive-set-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions are co-analytic and not Borel in the space of marked groups. We then adapt our techniques to show elementary amenable marked groups may be characterized by well-founded descriptive-set-theoretic trees,…

初等顺从群的凝聚性

Coherence for elementary amenable groups

该研究针对初等顺从群，证明群凝聚性、同调凝聚性与整群环凝聚性等价，推广了Bieri和Strebel关于有限生成可解群的相关结论。

The Asymptotic Dimension of Box Spaces for Elementary Amenable Groups

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.

某些区间交换变换的初等可容许子群

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。

A quasi-isometric embedding theorem for groups

We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups.

Ergodic amenable actions of algebraic groups

We prove that every ergodic amenable action of an algebraic group over a local field of characteristic zero is induced from an ergodic action of an amenable subgroup.

Algebraic entropy for amenable semigroup actions

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case $S=\mathbb N$. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a B…

Homological Dimension of Solvable Groups

In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.

The Strong Approximation Conjecture holds for amenable groups

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.

Some remarks on the entropy for algebraic actions of amenable groups

In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to infinity. These results extend previous work by Lind and Ward on p-adic entropy.

A non-LEA Sofic Group

We describe elementary examples of finitely presented sofic groups which are not residually amenable (and thus not initially subamenable or LEA, for short). We ask if an amalgam of two amenable groups over a finite subgroup is residually amenable and answer this positively for some special cases, including countable locally finite groups, residually nilpotent groups and others.

Amenable hyperbolic groups

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic loca…

关于李群和代数群中冗余生成集最大尺寸的探讨

On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups

研究显示，连通紧致李群的生成集若大小超过群秩的固定多项式，则必冗余。类似结果适用于可解李群和具有 Zariski 群的半单代数群。方法产生的量化界限受有限简单李型群界限的控制，并部分回答了 Gelander 的猜想。

The $L^2$-torsion polytope of amenable groups

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.

Algebraic entropy of amenable group actions

Let $R$ be a ring, let $G$ be an amenable group and let $R\ast G$ be a crossed product. The goal of this paper is to construct, starting with a suitable additive function $L$ on the category of left modules over $R$, an additive function on a subcategory of the category of left modules over $R\ast G$, which coincides with the whole category if $L({}_RR) <\infty$. This construction can be performed using a dynamical invariant associated with the original function $L$, called algebraic $L$-entropy. We apply our results to two classical problems on group rings: the Stable Finiteness and the Zero-…

Operator algebraic approach to inverse and stability theorems for amenable groups

We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.

Non-amenability of product replacement graphs

We prove non-amenability of the product replacement graphs Γ_n(G) for uniformly non-amenable groups. We also prove it for Z-large groups, when n is sufficiently large. It follows that Γ_n(G) is non-amenable when n is sufficiently large for hyperbolic groups, linear groups, and elementary amenable groups.

初等阿贝尔子群：从代数群到有限群

Elementary abelian subgroups: from algebraic groups to finite groups

本研究提出代数闭域上单代数群初等阿贝尔子群共轭类分类新方法，给出环面子群分类算法、例外型非环面子群结构表，并说明如何借助Lang-Steinberg定理迁移至有限李型群，为模表示论相关研究提供支撑。

A note on relative amenability

P-E. Caprace and N. Monod isolate the class $\mathscr{X}$ of locally compact groups for which relatively amenable closed subgroups are amenable. It is unknown if $\mathscr{X}$ is closed under group extension. In this note, we exhibit a large, group extension stable subclass of $\mathscr{X}$, which suggests $\mathscr{X}$ is indeed closed under group extension. Along the way, we produce generalizations of the class of elementary groups and obtain information on groups outside $\mathscr{X}$.

Maximal amenable subgroups of arithmetic groups

By classifying $S$-maximal amenable subgroups of algebraic groups over a global field of characteristic zero, we obtain a complete classification of maximal amenable subgroups up to commensurability in the respective arithmetic groups. Futhermore, we prove that these commensurably maximal amenable subgroups are singular and therefore give rise to maximal amenable von Neumann subalgebras.

Complexity among the finitely generated subgroups of Thompson's group

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…

顺从群何时具有实秩零$C^*$-代数

When amenable groups have real rank zero $C^*$-algebras

该研究探讨离散顺从群具有实秩零$C^*$-代数的条件，在局部有限群对应实秩零代数的已知结论基础上，证明了实秩零条件下初等顺从且有限Hirsch长度的正规子群必为局部有限群，推进了相关开放问题的研究。

Variations on a theme by Higman

We propose elementary and explicit presentations of groups that have no amenable quotients and yet are SQ-universal. Examples include groups with a finite classifying space, no Kazhdan subgroups and no Haagerup quotients.

Elementary abelian 2 subgroups of compact Lie groups

We classify elementary abelian 2 subgroups of compact simple Lie groups of adjoint type. This finishes the classification of elementary abelian $p$ subgroups of compact (or linear algebraic) simple groups of adjoint type.

Almost algebraic actions of algebraic groups and applications to algebraic representations

Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon due to Bader and Furman.

从望远镜到框架与单群

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。

一个有限生成的ICC非性质(T)群，既非内可均也非适当近端

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。

Iterated Monodromy Groups of Exponential Maps

This paper introduces iterated monodromy groups for transcendental functions and discusses them in the simplest setting, for post-singularly finite exponential functions. These groups are self-similar groups in a natural way, based on an explicit construction in terms of kneading sequences. We investigate the group theoretic properties of these groups, and show in particular that they are amenable, but they are not elementary subexponentially amenable.

代数廉价重建性质

The algebraic cheap rebuilding property

本文提出公理化框架处理群和链复形的同调性质组合定理，引入代数廉价重建性质，证明某些图群挠同调增长消失。

Fixed points and amenability in non-positive curvature

Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the torsion-free case. We establish Levi decompositions for stabilisers of points at infinity of X, generalising the case of linear algebraic groups to Is(X). A geometric counterpart of this sheds light on the refined bordification of X (à la Karpelevich) and leads to a converse to the Adams-Ballmann theorem. It is further deduced that unimodular cocompact groups cannot …

Morita equivalence classes of blocks with elementary abelian defect groups of order 16

We classify the Morita equivalence classes of blocks with elementary abelian defect groups of order $16$ with respect to a complete discrete valuation ring with algebraically closed residue field of characteristic two. As a consequence, blocks with this defect group are derived equivalent to their Brauer correspondent in the normalizer of a defect group and so satisfy Broué's Conjecture.

Ergodic theorems for affine actions of amenable groups on Hilbert space

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…

The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups

We show that if $H$ is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group $G$ then the Schreier coset graph $X$ for $G$ relative to $H$ is non-amenable (that is, $X$ has positive Cheeger constant). We present some corollaries regading the Martin boundary and Martin compactification of $X$ and the co-growth of $H$ in $G$.

Self-avoiding walks and amenability

The connective constant $μ(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called 'graph height functions', namely: (i) whether $μ(G)$ is a local function on certain graphs derived from $G$, (ii) the equality of $μ(G)$ and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating $μ(G)$ to a give…

顺从代数的线性 sofic 表示

Linear sofic representations of amenable algebras

本文研究代数的线性 sofic 近似概念，证明对于无零因子的有限生成顺从 K-代数，所有线性 sofic 表示共轭，并应用于秩度量下的弱稳定性问题。

On abstract representations of the groups of rational points of algebraic groups and their deformations

In this paper, we continue our study of abstract representations of elementary subgroups of Chevalley groups of rank $\geq 2.$ First, we extend our earlier methods to analyze representations of elementary groups over arbitrary associative rings, and as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form ${\bf SL}_{n,D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic zero. Second, we apply our results to study deformations of representations of elem…

An Elementary Linear-Algebraic Proof without Computer-Aided Arguments for the Group Law on Elliptic Curves

The group structure on the rational points of elliptic curves plays several important roles, in mathematics and recently also in other areas such as cryptography. However, the famous proofs for the group property (in particular, for its associative law) require somewhat advanced mathematics and therefore are not easily accessible by non-mathematician. On the other hand, there have been attempts in the literature to give an elementary proof, but those rely on computer-aided calculation for some part in their proofs. In this paper, we give a self-contained proof of the associative law for this o…

Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.

Lie、群与Hopf代数的可均性

Amenability of Lie, Group and Hopf algebras

本文通过泛包络代数中几乎不变的有限维子余代数定义Lie代数可均性，证明其与Elek定义等价，并推广至Hopf代数，验证Gromov猜想及多种封闭性质。

弱顺从群的Bernoulli作用的唯一Cartan子代数结果

A unique Cartan subalgebra result for Bernoulli actions of weakly amenable groups

该研究针对弱顺从群的Bernoulli作用相关算子代数问题，通过结合两类群的刚性结果，证明了满足特定条件的群作用对应的交叉积同构可推出其Cartan子代数酉共轭。

Isoperimetric profile and random walks on locally compact solvable groups

We study a large class of amenable locally compact groups containing all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties which are likely to be only satisfied by these groups.

Quasi-Hermitian locally compact groups are amenable

A locally compact group $G$ is called Hermitian if the spectrum $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in L^1(G)$ satisfying $f=f^*$, and called quasi-Hermitian if $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in C_c(G)$ satisfying $f=f^*$. We show that every quasi-Hermitian locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of "spectral interpolation of triple Banach $*$-algebr…

关于顺从的 Hilbert–Schmidt 稳定群

On amenable Hilbert-Schmidt stable groups

本文研究离散顺从群的 HS-稳定性，证明幂零群情形及中心扩张、半直积中的保持与刻画，并构造首个非置换稳定的例子，还厘清灵活稳定性与极大殆周期性的等价关系。

Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups

In this version small mistakes are corrected and the exposition is changed as suggested by the referee (to appear in Canadian Journal of Mathematics). The first main result of the paper is a criterion for a partially commutative group $\GG$ to be a domain. It allows us to reduce the study of algebraic sets over $\GG$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\GG$ (of a coordinate group over $\GG$) to the elementary theories of the direct factors of $\GG$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish norma…

Elementary subgroup of an isotropic reductive group is perfect

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that the elementary subgroup E(R) of group of points G(R) is correctly defined. Then E(R) is perfect, except for the well-known cases of a split reductive group of type C_2 or G_2.

An obstruction to $\ell^p$-dimension

For any group $G$ containing an infinite elementary amenable subgroup, and any $2<p<\infty$, there exists closed invariant subspaces $E_i\nearrow \ell^pG$ and $F\neq 0$ such that $E_i\cap F = 0$ for all $i$. This is an obstacle to $\ell^p$-dimension and gives a negative answer to a question of Gaboriau.

Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64

We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.

Mean Dimension, Mean Rank, and von Neumann-Lück Rank

We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $Γ$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $Γ$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Lück rank of M all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups…

初等群的外幂的上群：正规化子

Overgroups of exterior powers of an elementary group. Normalizers

本文刻画了代数群概型 $\bigwedge^m GL_n$ 的几何与代数性质，并将其作为正规化子应用于初等子群上群的“三明治分类”及线性保持问题。

Properly proximal groups and their von Neumann algebras

We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first $W^*$-strong rigidity results for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.

多项式向量场初等可积性的约化

Reduction of Elementary Integrability of Polynomial Vector Fields

本文推广 Prelle-Singer 结果，证明任意 $n$ 维初等可积多项式向量场具有 $n-1$ 个由代数元素构成的函数独立第一积分，并借助 Galois 理论得到有理 Jacobian 乘子。