The vanishing-off subgroup

Mixing, malnormal subgroups and cohomology in degree one

The aim of the current paper is to explore the implications on the group $G$ of the non-vanishing of the cohomology in degree one of one of its representation $π$, given some mixing conditions on $π$. In one direction, harmonic cocycles are used to show that the FC-centre should be finite (for mildly mixing unitary representations). Next, for any subgroup $H<G$, $H$ will either be "small", almost-malnormal or $π_{|H}$ also has non-trivial cohomology in degree one (in this statement, "small", reduced vs unreduced cohomology and unitary vs generic depend on the mixing condition). The notion of q…

Weak second maximal subgroups in solvable groups

In this paper, we investigate the difference between weak second maximal subgroups and second maximal subgroups. A sufficient and necessary condition is also given to describe such class of groups whose weak second maximal subgroups coincide with its second maximal subgroups(called WSM-groups) under the solvable case. As an application, we show that every non-vanishing element of a solvable WSM-group lies in its Fitting subgroup.

素幂阶的消失元素及其类大小性质

Vanishing elements of prime power order and their class size property

该研究在Dolfi和Lucido的工作基础上引入了群的性质$P_v(p,q)$，证明有限单群不满足该性质，并推广了相关可解性结果。

On the second homology group of the Torelli subgroup of Aut(F_n)

Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H_2(IA_n) as a GL_n(Z)-module. Corollaries include a version of surjective representation stability for H_2(IA_n), the vanishing of the GL_n(Z)-coinvariants of H_2(IA_n), and the vanishing of the second rational homology group of the level l congruence subgroup of Aut(F_n). Our generating set is derived from a new group presentation for IA_n which is infinite but which has a simple recursive form.

$L^2$-Betti numbers and non-unitarizable groups without free subgroups

We show that there exist non-unitarizable groups without non-abelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely generated torsion groups with non-vanishing first $L^2$-Betti numbers. We also relate the well-known problem of whether every hyperbolic group is residually finite to an open question about approximation of $L^2$-Betti numbers.

互素作用与不动点子群上非零的特征标

Coprime Actions and Characters Non-vanishing on the Fixed-point Subgroup

本文否定回答了 Navarro 关于互素作用下不变特征标在不动点子群上非零个数的猜想，通过构造最小反例并给出机器验证。

Weak notions of normality and vanishing up to rank in L2-cohomology

We study vanishing results for L2-cohomology of countable groups under the presence of subgroups that satisfy some weak normality condition. As a consequence we show that the L2-Betti numbers of SL(n,R) for any infinite integral domain R vanish below degree n-1. We also give a uniform proof for the vanishing of L2-Betti numbers of Thompson's groups F and T.

The Bogomolov multiplier of rigid finite groups

The Bogomolov multiplier of a finite group $G$ is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of $G$. This invariant of $G$ plays an important role in birational geometry of quotient spaces $V/G$. We show that in many cases the vanishing of the Bogomolov multiplier is guaranteed by the rigidity of $G$ in the sense that it has no outer class-preserving automorphisms.

On characters of Chevalley groups vanishing at the non-semisimple elements

Let G be a finite simple group of Lie type. In this paper we study characters of G that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of G. Such characters can be viewed as a natural generalization of the Steinberg character. For groups G of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.

On zeros of irreducible characters lying in a normal subgroup

Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $χ(g)\neq 0$ for all irreducible characters $χ$ of $G$. Such an element is said to be non-vanishing in $G$. Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property, then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.

关于群的$L^2$-Betti数消失准则

On vanishing criteria of $L^2$-Betti numbers of groups

本文推广了群的$L^2$-Betti数消失准则，利用无限指数子群及其所含正规子群的$L^2$-Betti数条件证明$G$的$L^2$-Betti数在给定次数以下消失，并给出Gaboriau定理的纯代数证明及Hillman问题的肯定证据。

素数幂阶的消失元素

Vanishing Elements of Prime Power Order

本文证明了除 $\mathrm{SL}_2(4)$ 和 $\mathrm{SL}_2(8)$ 外的非交换有限单群均含共轭类大小被三个不同素数整除的素数幂阶消失元素，并据此推广了Robati关于非可解有限群的相关结论。

Trivial source endo-trivial modules for finite groups with a semi-dihedral Sylow $2$-subgroup

We finish off the classification of the endo-trivial modules of finite groups with Sylow $2$-subgroups isomorphic to a semi-dihedral $2$-group started by Carlson, Mazza and Thévenaz in their article "Endotrivial modules over groups with quaternion or semi-dihedral Sylow $2$-subgroup" published in 2013.

The Howe-Moore property for real and p-adic groups

We consider in this paper a relative version of the Howe-Moore Property, about vanishing at infinity of coefficients of unitary representations. We characterize this property in terms of ergodic measure-preserving actions. We also characterize, for linear Lie groups or p-adic Lie groups, the pairs with the relative Howe-Moore Property with respect to a closed, normal subgroup. This involves, in one direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of …

Isaacs_Navarro_Wolf猜想的一个证明

A Proof of the Isaacs_Navarro_Wolf Conjecture

本文证明了有限可解群的非零元必属于Fitting子群，通过结合Isaacs等人的归约与有限辛空间、Clifford理论完成了Isaacs_Navarro_Wolf猜想的证明。

A generalization of the Lyndon--Hochschild--Serre spectral sequence with applications to group cohomology and decompositions of groups

We set up a Grothendieck spectral sequence which generalizes the Lyndon--Hochschild--Serre spectral sequence for a group extension $K\mono G\epi Q$ by allowing the normal subgroup $K$ to be replaced by a subgroup, or family of subgroups which satisfy a weaker condition than normality. This is applied to establish a decomposition theorem for certain groups as fundamental groups of graphs of Poincaré duality groups. We further illustrate the method by proving a cohomological vanishing theorem which applies for example to Thompson's group $F$.

Structure of the Group of Balanced Labelings on Graphs, its Subgroups and Quotient Groups

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two closely related to it groups: the subgroup of balanced labelings which consists of functions vanishing on vertices and the corresponding factor-group. This work is completely self-contained, except the algorithm for obtaining the 3-edge-connected components of an undirected graph, for which we…

On solvable groups with one vanishing class size

Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$ whose elements are the conjugacy class sizes of the vanishing elements of $G$. Motivated by the results in \cite{BLP}, we describe the class of the finite groups $G$ such that vcs$(G)$ consists of a single element \emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylo…

On the orders of vanishing elements of finite groups

Let $ G$ be a finite group and $p$ be a prime. Let $ \mathrm{Vo}(G) $ denote the set of the orders of vanishing elements, $\mathrm{Vo}_{p} (G)$ be the subset of $ \mathrm{Vo}(G) $ consisting of those orders of vanishing elements divisible by $p$ and $\mathrm{Vo}_{p'} (G) $ be the subset of $ \mathrm{Vo}(G) $ consisting of those orders of vanishing elements not divisible by $p$. Dolfi, Pacifi, Sanus and Spiga proved that if $ a $ is not a $ p $-power for all $ a\in \mathrm{Vo}(G)$, then $ G $ has a normal Sylow $ p $-subgroup. In another article, the same authors also show that if if $ \mathrm{…

Reading Off Kurosh Decompositions

Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used by many authors to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ our generalized Stallings' folding method to introduce a procedure, which given a subgroup H of a free product of finite groups reads off its Kurosh decomposition from the subgroup graph of H.

Bogomolov multipliers for unitriangular groups

The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of $G$. In this paper we give a positive answer to an open problem posed by Kang and Kunyavskiĭ in \cite{KK}. Namely, we prove that if $G$ is either a unitriangular group over $\f$, a quotient of its lower central series, a subgroup of its lower central series, or a central product of two unitriangular groups, then $B_0(G)=0$.

Herzog-Schonheim猜想、单位根的消没和与凸多边形

Herzog-Schonheim conjecture, vanishing sums of roots of unity and convex polygons

本文将Herzog-Schönheim猜想转化为单位根消没和与凸多边形问题，并利用该新方法证明了一些相关结果。

特征标、Hall子群与正规补群

Characters, Hall subgroups, and Normal Complements

本文在更一般假设下，为1989年关于有限群Hall子群的不可约复特征标延拓与正规补群等价的经典结果提供了模版本。

Vanishing of l^2-cohomology as a computational problem

We show that it is impossible to algorithmically decide if the l^2-cohomology of the universal cover of a finite CW complex is trivial, even if we only consider complexes whose fundamental group is equal to the elementary amenable group (Z_2 \wr Z)^3. A corollary of the proof is that there is no algorithm which decides if an element of the integral group ring of the group (\Z_2 \wr Z)^4 is a zero-divisor. On the other hand, we show, assuming some standard conjectures, that such an algorithm exists for the integral group ring of any group with a decidable word problem and a bound on the sizes o…

结群的分支子群及其 profinite 和上同调结构

Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure

本文研究结群的分支子群及其 profinite 和上同调结构，通过定义meridional inertia子群和全局分支子群，分析其在有限商、profinite 完成和上同调中的性质。

Groups having all elements off a normal subgroup with prime power order

We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either $G$ is a $p$-group or $G = PN$ where $P$ is a Sylow $p$-subgroup and $(G,P,P \cap N)$ is a Frobenius-Wielandt triple. We also prove that if all the elements of $G \setminus N$ have prime power orders and the orders are divisible by two primes $p$ and $q$, then $G$ is a $\{ p, q \}$-group and $G/N$ is either a Frobenius group or a $2$-Frobenius group. If all th…

On the top-dimensional $\ell^2$-Betti numbers

The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional $\ell^2$-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the $\ell^2$-Betti numbers of Aut($F_n$) and Out($F_n$) (and of their Torelli subgroups) do not vanish in degree equal to their virtual cohomological dimension, we prove that the subgroups of the 3-manifold groups have vanishing $\ell^2$-Betti numbers in degree 3 and 2 and we prove for instance that $F_2^d \times Z$ has ergodic dimension $d + 1$.

Computing the order and the index of a subgroup in a polycyclic group

This contains a new version of the so-called non-commutative Gauss algorithm for polycyclic groups. Its results allow to read off the order and the index of a subgroup in an (possibly infinite) polycyclic group.

A note on approximate subgroups of GL_n(C) and uniformly nonamenable groups

The aim of this brief note is to offer another proof of a theorem of Hrushovski that approximate subgroups of GL_n(C) are almost nilpotent. This approach generalizes to uniformly non amenable groups.

Random presentations and random subgroups: a survey

This is a survey of results on random group presentations, and on random subgroups of certain fixed groups. Being a survey, this paper does not contain new results, but it offers a synthetic view of a part of this very active field of research.

On the Skitovich-Darmois theorem for some locally compact Abelian groups

Let $X$ be a locally compact Abelian group, $α_{j}, β_j$ be topological automorphisms of $X$. Let $ξ_1, ξ_2$ be independent random variables with values in $X$ and distributions $μ_j$ with non-vanishing characteristic functions. It is known that if $X$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$, then the independence of the linear forms $L_1=α_1ξ_1+α_2ξ_2$ and $L_2=β_1ξ_1+β_2ξ_2$ implies that $μ_j$ are Gaussian distributions. We prove that if $X$ contains no subgroup topologically isomorphic to $\mathbb{T}^2$, then the independence of $L_1$ and $L_2$ implies…

2-subnormal quadratic offenders and Oliver's p-group conjecture

Bob Oliver conjectures that if $p$ is an odd prime and $S$ is a finite $p$-group, then the Oliver subgroup $\X(S)$ contains the Thompson subgroup $J_e(S)$. A positive resolution of this conjecture would give the existence and uniqueness of centric linking systems for fusion systems at odd primes. Using ideas and work of Glauberman, we prove that if $p \geq 5$, $G$ is a finite $p$-group, and $V$ is an elementary abelian $p$-group which is an F-module for $G$, then there exists a quadratic offender which is 2-subnormal (normal in its normal closure) in $G$. We apply this to show that Oliver's co…

Uniform rank gradient, cost and local-global convergence

We analyze the rank gradient of finitely generated groups with respect to sequences of subgroups of finite index that do not necessarily form a chain, by connecting it to the cost of p.m.p. actions. We generalize several results that were only known for chains before. The connection is made by the notion of local-global convergence. In particular, we show that for a finitely generated group $Γ$ with fixed price $c$, every Farber sequence has rank gradient $c-1$. By adapting Lackenby's trichotomy theorem to this setting, we also show that in a finitely presented amenable group, every sequence o…

Hecke algebras for protonormal groups

We introduce the term "protonormal" to refer to a subgroup H of a group G such that for every x in G the subgroups x^{-1}Hx and H commute as sets. If moreover (G,H) is a Hecke pair we show that the Hecke algebra H(G,H) is generated by the range of a canonical partial representation of G vanishing on H. As a consequence we show that there exists a maximum C*-norm on H(G,H), generalizing previous results by Brenken, Hall, Laca, Larsen, Kaliszewski, Landstad and Quigg. When there exists a normal subgroup N of G, containing H as a normal subgroup, we prove a new formula for the product of the gene…

Rank, combinatorial cost and homology torsion growth in higher rank lattices

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…

柱形双曲群中的拟凸子群概念

A notion of quasiconvex subgroups in acylindrically hyperbolic groups

本文在含双曲嵌入子群的有限生成群中提出拟凸性概念，证明其在锥化尖点空间有一致拟凸常数、长Dehn填充下保持不变，并推广了Groves-Manning关于CAT(0)立方复形群作用的定理。

关于$p$进解析群作用上同调的一则注记

A note on the cohomology of $p$-adic analytic group actions

本文研究紧$p$进李群解析作用的连续上同调，证明可通过pro-$p$ pro循环子群的对应性质检测其上同调的同时消失或有限维性，并提出推广猜想及李代数上同调的相关猜想。

Existence, uniqueness and functoriality of the perfect locality over a Frobenius P-category

Let p be a prime, P a finite p-group and F a Frobenius P-category. The question on the existence of a suitable category Lsc extending the full subcategory of F over the set of F-selfcentralizing subgroups of P goes back to Dave Benson in 1994. In 2002 Carles Broto, Ran Levi and Bob Oliver formulate the existence and the uniqueness of the category Lsc in terms of the annulation of an obstruction 3-cohomology element and of the vanishing of a 2-cohomology group, and they state a sufficient condition for the vanishing of these n-cohomology groups. Recently, Amy Chermak has proved the existence an…

唯一性阈值处的渗流：通过子群相对化方法

Percolation at the uniqueness threshold via subgroup relativization

本文通过子群相对化方法，在包含指数增长可遗 $wq$-正规子群的群类中证明了唯一性阈值处的非唯一性，并建立了强非唯一性的共遗传性质，同时解决了随机游走与渗流簇相交的公开问题。

有界上同调消失的代数判别准则

An algebraic criterion for the vanishing of bounded cohomology

本文提出一个称为交换循环共轭条件的代数判别准则，证明了许多几何、动力和代数中重要群（如保结构微分同胚群、区间交换变换群、分段线性/射影群、直极限线性群及大映射类群的子群）的可分对偶系数有界上同调消失，并证明了Bowden猜想。

完全约化与例外代数群的子群

Complete reducibility and subgroups of exceptional algebraic groups

本文综述了Serre的$G$-完全约化概念及其在阐明例外型单代数群子群结构中的推论，并概述了该领域六十年来的已知成果及文献更正。

Coxeter群的双曲模型与稳定性识别

Hyperbolic models and stability recognition for Coxeter groups

本文针对有限秩Coxeter系统，通过锥化Davis复形的宽抛物子复形构造双曲空间，证明该空间兼具稳定性识别与Morse识别性质，且Coxeter群在其上的作用给出其最大非阿基米德双曲结构。

Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the…

Gaussian Binomial Coefficients in Group Theory, Field Theory, and Topology

In this article, we offer group-theoretic, field-theoretic, and topological interpretations of the Gaussian binomial coefficients and their sum. For a finite $p$-group $G$ of rank $n$, we show that the Gaussian binomial coefficient $\binom{n}{k}_p$ is the number of subgroups of $G$ that are minimally expressible as an intersection of $n - k$ maximal subgroups of $G$, and their sum is precisely the number of subgroups that are either $G$ or an intersection of maximal subgroups of $G$. We provide a field-theoretic interpretation of these quantities through the lens of Galois theory and a topolog…

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian …