Discrete groups, Grothendieck rings and families of finite subgroups

The true prosoluble completion of a group: examples and open problems

The true prosoluble completion $P\Cal S (Γ)$ of a group $Γ$ is the inverse limit of the projective system of soluble quotients of $Γ$. Our purpose is to describe examples and to point out some natural open problems. We discuss a question of Grothendieck for profinite completions and its analogue for true prosoluble and true pronilpotent completions. 1. Introduction. 2. Completion with respect to a directed set of normal subgroups. 3. Universal property. 4. Examples of directed sets of normal subgroups. 5. True prosoluble completions. 6. Examples. 7. On the true prosoluble and the true pronilpo…

On the dimension of classifying spaces for families of abelian subgroups

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 $r\geq 0$.

Euler classes and Bredon cohomology for groups with restricted families of finite subgroups

We explore some of the special features with respect to Bredon cohomology of groups having all its finite subgroups either nilpotent or p-groups or cyclic p-groups. We get some results on dimensions and also a formula for the equivariant Euler class for certain groups. We consider the generalization for Bredon cohomology of the properties of being duality or Poincaré duality and study their behavior under p-power index extensions with coefficients in a field of characteristic p.

相对双曲群、格罗滕迪克对与纤维积中不可数的投射有限歧义

Relatively hyperbolic groups, Grothendieck pairs, and uncountable profinite ambiguity among fibre products

本文研究相对双曲群直积的投射有限刚性丧失问题，提出构造无限格罗滕迪克对和不可数互不同构子群族的一般方法，并以Weeks流形基本群和8字结补4重分支覆盖基本群为例，证明格罗滕迪克对完全解释了其直积的刚性丧失。

子群族的分类空间的有界上同调

Bounded cohomology of classifying spaces for families of subgroups

本文引入子群族的有界 Bredon 上同调，推广有界上同调，并给出相对可遗性与相对双曲性的上同调刻画。

Classifying spaces for chains of families of subgroups

This thesis concerns the study of the Bredon cohomological and geometric dimensions of a discrete group $G$ with respect to a family $\mathfrak{F}$ of subgroups of $G$. With that purpose, we focus on building finite-dimensional models for $\operatorname{E}_{\mathfrak{F}} \left( G \right)$. The cases of the family $\mathfrak{Fin}$ of finite subgroups of a group and the family $\mathfrak{VC}$ of virtually cyclic subgroups of a group have been widely studied and many tools have been developed to relate the classifying spaces for $\mathfrak{VC}$ with those for $\mathfrak{Fin}$. Given a discrete gr…

Saturated fusion systems with parabolic families

Let G be group; a finite p-subgroup S of G is a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S. In this paper, we examine the relations between the fusion system over S which is given by conjugation in G and a certain chamber system C, on which G acts chamber transitively with chamber stabilizer N_G(S). Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underly…

Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, $Out(F_n)$, and the Cremona group. Other examples can be found among groups acting geometrically on $CAT(0)$…

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 $F$.

On families of nilpotent subgroups and associated coset posets

We study some properties of the coset poset associated with the family of subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is $2$-Engel, and $2$-connected if and only if the group is nilpotent of class $2$ or less. We determine the homotopy type of the coset poset for the group of $4\times 4$ upper unitriangular matrices over $\mathbb{F}_p$, and for the Burnside groups of exponent $3$.

Linear Groups, Conjugacy Growth, and Classifying Spaces for Families of Subgroups

Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its classifying space $E_{\mathcal F}G$ with respect to $\mathcal{F}$. When $\mathcal F = \mathcal{VC}yc$ is the family of virtually cyclic subgroups, Juan-Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by Lück--Reich--Rognes--Varisco for the family of cyclic subgroups. Finally, we constru…

Groups acting on trees and the Eilenberg-Ganea problem for families

We construct new examples of groups with cohomological dimension 2 and geometric dimension 3 with respect to the families of finite subgroups, virtually abelian groups of bounded rank, and the family of virtually poly-cyclic subgroups. Our main ingredients are the examples constructed by Brady-Leary-Nucinckis and Fluch-Leary, and Bass-Serre theory.

On independent families of normal subgroups in free groups

Consider a presentation $\mathcal{P}=<{\bf x}\mid{\bf \bigcup_{i=1}^n r_i}>$. Let ${\bf R_i}$ be the normal closure of the set ${\bf r_i}$ in the free group ${\bf F}$ with basis ${\bf x}$, $\mathcal{P}_i=<{\bf x}\mid{\bf r_i}>$, ${\bf N_i} = \prod_{j\neq i}{\bf R_j}$. In the present article, using geometric techniques of pictures, generators for $\frac{{\bf R_i}\cap {\bf N_i}}{[{\bf R_i}, {\bf N_i}]}$, $i=1,...,n$, are obtained from a set of generators over $\{\mathcal{P}_i\mid i=1,..., n\}$ for $π_2(\mathcal{P})$. As a corollary, we get a sufficient condition for the family $\{{\bf R_1},...,{…

分支群的弱极大子群

Weakly maximal subgroups of branch groups

本文研究分支群的弱极大子群，描述了两个族（射线稳定化子的推广及块结构族），并证明对Grigorchuk群和挠性GGS群所有弱极大子群均属此二族，给出多种等价刻画。

运动规划不变量与子群族

Motion planning invariants and families of subgroups

本文引入关于子群族的拓扑复杂度不变量，提出置换族并证明其下界性质，从而推广了经典与分布拓扑复杂度相等的结果，并证明Farber猜想在分布意义上成立。

Subgroup families controlling p-local finite groups

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at …

Higher generating subgroups and Cohen-Macaulay complexes

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi-subgroups and describe higher generating families of parabolic subgroups in Aut(F_n).

Nilpotent completions of groups, Grothendieck pairs, and four problems of Baumslag

Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-fre…

Some results related to finiteness properties of groups for families of subgroups

For a group $G$ we consider the classifying space $E_{\mathcal{VC}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{\mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of Lück-Reich-Rognes-Varisco for Artin groups. We then study the conjugacy growth of CAT(0) groups and show that if a CAT(0) group contains a free abelian group of rank two, its conjugacy growth is strictly faster than linear. This also yields an alternative proof for the fact that a CAT(0) cube g…

Classifying spaces for families of subgroups for systolic groups

We determine the large scale geometry of the minimal displacement set of a hyperbolic isometry of a systolic complex. As a consequence, we describe the centraliser of such an isometry in a systolic group. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on a systolic complex. Its dimension coincides with the topological dimension of the complex if the latter is at least four. We show that graphical small cancellation complexes are classifying spaces for proper actions and that the groups acting on them…

基于非负欧拉特征表面纯辫群虚阿贝尔子群族的分类空间

Classifying spaces for families of virtually abelian subgroups of surface braid groups

本文研究了表面纯辫群虚阿贝尔子群族的分类空间的最小维数，证明其等于群的虚上同调维数加n，并应用于计算可算子群族的分类空间最小维数。

具有环面子群格的有限阿贝尔群

Finite Abelian Groups with Toroidal Subgroup Lattices

研究确定了多类阿贝尔群的子群格亏格，完成了子群格可嵌入环面的全部有限阿贝尔群的分类工作。

离散亨泽尔赋值环上Bruhat-Tits子群的算术

Arithmétique des sous-groupes de Bruhat-Tits sur un anneau de valuation discrète hensélien

本文细化了Bruhat-Tits子群的上同调研究，补充1987年工作，并对半单单连通群得到Grothendieck-Serre型结果及拟分裂伴随群情形的障碍精确计算。

Universal family of the subgroups of an algebraic group

We construct a moduli space for the connected subgroups of an algebraic group and the corresponding universal family. Morphisms from an algebraic variety to this moduli space correspond to flat families of connected algebraic subgroups parametrised by this variety. This moduli space is obtained by gluing together infinitely many irreducible projective varieties of bounded dimension along closed subvarieties. Regarding families of non-connected subgroups of an algebraic group, we show that, given sich a family, the corresponding family of identity components is an irreducible component of the f…

Grothendieck-Teichmueller群温和版本的GT-shadows

GT-shadows for the gentle version of the Grothendieck-Teichmueller group

本研究针对Grothendieck-Teichmueller群的温和版本$GT_{gen}$构建了GT-shadows群胚$GTSh$，证明$GT_{gen}$同构于相关函子的极限，并给出识别真正GT-shadows的判别准则。

Coset Posets of Infinite Groups

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index, and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

Brauer Theory for Profinite Groups

Brauer Theory for a finite group can be viewed as a method for comparing the representations of the group in characteristic 0 with those in prime characteristic. Here we generalize much of the machinery of Brauer theory to the setting of profinite groups. By regarding Grothendieck groups as functors we describe corresponding Grothendieck groups for profinite groups, and generalize the decomposition map, regarded as a natural transformation. We discuss characters and Brauer characters for profinite groups. We give a functorial description of the block theory of a profinite group. We finish with…

Maximal subgroups and irreducible representations of generalised multi-edge spinal groups

Let $p\ge 3$ be a prime. A generalised multi-edge spinal group is a subgroup of the automorphism group of a regular $p$-adic rooted tree T that is generated by one rooted automorphism and $p$ families of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalises the concepts of multi-edge spinal groups, including the widely studied GGS-groups, and the extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised multi-edge spi…

Decompositions of congruence subgroups of Chevalley groups

We formulate and prove relative versions of several classical decompositions known in the theory of Chevalley groups over commutative rings. As an application we obtain upper estimates for the width of principal congruence subgroups in terms of several families of generators. Some of our results are new even in the absolute case and were previously studied only for groups over finite fields.

Obstructions for subgroups of Thompson's group $V$

Thompson's group $V$ has a rich variety of subgroups, containing all finite groups, all finitely generated free groups and all finitely generated abelian groups, the finitary permutation group of a countable set, as well as many wreath products and other families of groups. Here, we describe some obstructions for a given group to be a subgroup of $V$.

关于允许收缩到抛物子群的Artin群

On Artin groups admitting retractions to parabolic subgroups

本文将偶Artin群的收缩推广至FC型及更广族，证明其唯一扩展至任意抛物子群，并据此推广相交问题的简化条件，刻画FC情形的凝聚性。

On normal subgroups of the braided Thompson groups

We inspect the normal subgroup structure of the braided Thompson groups Vbr and Fbr. We prove that every proper normal subgroup of Vbr lies in the kernel of the natural quotient Vbr \onto V, and we exhibit some families of interesting such normal subgroups. For Fbr, we prove that for any normal subgroup N of Fbr, either N is contained in the kernel of Fbr \onto F, or else N contains [Fbr,Fbr]. We also compute the Bieri-Neumann-Strebel invariant Sigma^1(Fbr), which is a useful tool for understanding normal subgroups containing the commutator subgroup.

On the Andreadakis equality for a subgroup of the McCool group

The McCool group has families of subgroups such as the ordinary pure braid group, the upper triangular McCool group and the partial inner automorphism group. The generalized Andreadakis conjecture holds for the ordinary pure braid group and the upper triangular McCool group. In this paper, we prove a similar result for the partial inner automorphism group and also establish an isomorphism between this group and the inner automorphism group of the upper triangular McCool group.

On the intersection of subgroups in free groups: echelon subgroups are inert

A subgroup $H$ of a free group $F$ is called inert in $F$ if for every $G < F$ the rank of the intersection of $H$ with $G$ is no grater than the rank of $G$. In this paper we expand the known families of inert subgroups. We show that the inertia property holds for 1-generator endomorphisms. Equivalently, echelon subgroups in free groups are inert. An echelon subgroup is defined through a set of generators that are in echelon form with respect to some ordered basis of the free group, and may be seen as a generalization of a free factor. For example, the fixed subgroups of automorphisms of fini…

On groups acting on contractible spaces with stabilizers of prime power order

We study actions of discrete groups on contractible topological spaces in which either (1) all stabilizers lie in the family of subgroups of prime power order or (2) all stabilizers lie in the family of finite subgroups. We compare the classifying spaces for actions with stabilizers in these two families, the Kropholler hierarchies build on these two families, and group cohomology relative to these two families.

Representations of braid groups via conjugation actions on congruence subgroups

We construct two families of representations of the braid group $B_n$ by considering conjugation actions on congruence subgroups of $GL_{n-1}(Z[t^{\pm 1},q^{\pm 1}])$. We show that many of these representations are faithful modulo the center of $B_n$.

Pink-type results for general subgroup of $\operatorname{SL}_2(\mathbb{Z}_\ell)^n$

We study open subgroups $G$ of $\operatorname{SL}_2(\mathbb{Z}_\ell)^n$ in terms of some associated Lie algebras without assuming that $G$ is a pro-$\ell$ group, thereby extending a theorem of Pink. The result has applications to the study of families of Galois representations.

幂零群与映射类群的子曲面子群的同余子群性质

Congruence Subgroup Property for nilpotent groups and subsurface subgroups of Mapping Class Groups

本文证明有限型曲面映射类群中两类子群满足同余子群性质，并顺带给出 Ben-Ezra–Lubotzky 定理的初等证明。

Hyperbolic Groups with Finitely Presented Subgroups not of Type $F_3$

We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than $3$.

Homomorphisms to acylindrically hyperbolic groups I: Equationally noetherian groups and families

We study the set of homomorphisms from a fixed finitely generated group into a family of groups which are `uniformly acylindrically hyperbolic'. Our main results reduce this study to sets of homomorphisms which do not diverge in an appropriate sense. As an application, we prove that any relatively hyperbolic group with equationally noetherian peripheral subgroups is itself equationally noetherian.

Intersection properties of stable subgroups and bounded cohomology

We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to the ambient group.

通过算术代数几何技术保持同构的族：在广义Neukirch-Uchida定理中的应用

Families preserving isomorphisms via techniques in anabelian geometry: with an application to a generalized Neukirch-Uchida theorem

本文从算术代数几何视角推广保持自同构的族的概念，证明一大类profinit群的相关同构族源自内自同构，并将其应用得到ℓ-拟数域的广义Neukirch-Uchida定理。

辫群、RAAGs 和交换群群的族分类空间

Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups

本文研究群中秩至多 $n$ 的 virtually 交换子群族 $\mathcal{F}_n$，证明自由交换群关于 $\mathcal{F}_n$ 的 Bredon 上同调非平凡性，并计算辫群、RAAGs 及相关群分类空间的最小维度。

Dynamical properties of profinite actions

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($τ$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in gene…

A Classification of the Stable Type of $BG$

We give a classification of the $p$--local stable homotopy type of $BG$, where $G$ is a finite group, in purely algebraic terms. $BG$ is determined by conjugacy classes of homomorphisms from $p$--groups into $G$. This classification greatly simplifies if $G$ has a normal Sylow $p$--subgroup; the stable homotopy types then depends only on the Weyl group of the Sylow $p$--subgroup. If $G$ is cyclic mod $p$ then $BG$ determines $G$ up to isomorphism. The last class of groups is important because in an appropriate Grothendieck group $BG$ can be written as a unique linear combination of $BH$'s, whe…

Subgroups of finite index and the just infinite property

A residually finite (profinite) group $G$ is just infinite if every non-trivial (closed) normal subgroup of $G$ is of finite index. This paper considers the problem of determining whether a (closed) subgroup $H$ of a just infinite group is itself just infinite. If $G$ is not virtually abelian, we give a description of the just infinite property for normal subgroups in terms of maximal subgroups. In particular, we show that such a group $G$ is hereditarily just infinite if and only if all maximal subgroups of finite index are just infinite. We also obtain results for certain families of virtual…

Free products from spinning and rotating families

The far-reaching work of Dahmani-Guirardel-Osin and recent work of Clay-Mangahas-Margalit provide geometric approaches to the study of the normal closure of a subgroup (or a collection of subgroups)in an ambient group $G$. Their work gives conditions under which the normal closure in $G$ is a free product. In this paper we unify their results and simplify and significantly shorten the proof of the Dahmani-Guirardel-Osin theorem.

Two Families of Examples of Groups Acting on Trees with Nontrivial Quasi-Kernels

We introduce two families of examples of groups acting on trees, one consisting of group amalgamations and the other consisting of HNN-extensions, motivated by the problems of $C^*$-simplicity and unique trace property. Moreover, we prove that our examples are not inner amenable and identify a relatively large, simple, normal subgroup in each one.

Dehn functions and finiteness properties of subgroups of perturbed right-angled Artin groups

We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type $\mathrm{FP}_3$, and have exponential, or polynomial Dehn functions of prescribed degree.

树自同构的二生成元离散子群

Two-Generator Discrete and Faithful Subgroups of Tree Automorphisms

本文部分分类了三叉树自同构群中满足小几何量限制的二生成元离散子群，并放宽限制时构造了无限族群图。

弱分支群的非闭子群

Non-closed subgroups of weakly branch groups

针对作用于足够正则根树的弱分支群G，构造两类非闭子群连续族：一类含非ERF子群，另一类在额外假设下含ERF子群，均在G的完备或同余拓扑中不闭。

The subgroup commutativity degree of finite P-groups

The subgroup commutativity degree of a group G has been defined in [6] as the probability that two subgroups of G commute, or equivalently that the product of two subgroups is again a subgroup. Problem 4.3 of [6] asks whether there exist families of groups other than dihedral, quasi-dihedral or generalized quaternion (all of 2-power cardinality), whose subgroup commutativity degree tends to 0 as the size of the group tends to infinity. An affirmative answer to this question has been provided by Aivazidis [1, 2] for the family of projective special linear groups over fields of even characterist…

On finiteness of some verbal subgroups in profinite groups

Given a group word $w$ and a group $G$, the set of $w$-values in $G$ is denoted by $G_w$ and the verbal subgroup $w(G)$ is the one generated by $G_w$. In the present paper we consider profinite groups admitting a word $w$ such that the cardinality of $G_w$ is less than $2^{\aleph_0}$ and $w(G)$ is generated by finitely many $w$-values. For several families of words $w$ we show that under these assumptions $w(G)$ must be finite. Our results are related to the concept of conciseness of group words.

On the number of cyclic subgroups of a finite group

Let $G$ be a finite group and let $c(G)$ be the number of cyclic subgroups of $G$. We study the function $α(G) = c(G)/|G|$. We explore its basic properties and we point out a connection with the probability of commutation. For many families $\mathscr{F}$ of groups we characterize the groups $G \in \mathscr{F}$ for which $α(G)$ is maximal and we classify the groups $G$ for which $α(G) > 3/4$. We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality $α(G) = α(G/N)$ when $G/N$ is a symmetric group.