Hereditarily non-topologizable groups

Inverse system characterizations of the (hereditarily) just infinite property in profinite groups

We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are 'universal' in that all (hereditarily) just infinite profinite groups arise as limits of the specified form. This is a corrected and revised version of the article: 'Inverse system characterizations of the (hereditarily) just infinite property in profinite groups', Bull. LMS vol 44, 3 (2012) 413--425.

On (hereditarily) just infinite profinite groups that are not virtually pro-p

A profinite group G is just infinite if every non-trivial closed normal subgroup of G is of finite index, and hereditarily just infinite if every open subgroup is just infinite. Hereditarily just infinite profinite groups need not be virtually pro-p, as shown in a recent paper of Wilson. The same paper gives a criterion on an inverse system of finite groups that is sufficient to ensure the limit is either virtually abelian or hereditarily just infinite. We give criteria of a similar nature that characterise the just infinite and hereditarily just infinite properties under the assumption that G…

The lower rank of direct products of hereditarily just infinite groups

We determine the lower rank of the direct product of finitely many hereditarily just infinite profinite groups of finite lower rank.

On hereditarily just infinite profinite groups obtained via iterated wreath products

We study a generalisation of the family of non-(virtually pro-$p$) hereditarily just infinite profinite groups introduced by J.\! S.\! Wilson in 2010. We prove that this family contains groups of finite lower rank. We also show that many groups in this family are not topologically finitely presentable.

On asymptotically hereditarily aspherical groups

We undertake a systematic study of asymptotically hereditarily aspherical (AHA) groups - the class of groups introduced by Tadeusz Januszkiewicz and the second author as a tool for exhibiting exotic properties of systolic groups. We provide many new examples of AHA groups, also in high dimensions. We relate AHA property with the topology at infinity of a group, and deduce in this way some new properties of (weakly) systolic groups. We also exhibit an interesting property of boundary at infinity for few classes of AHA groups.

On hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum

We prove that the inverse limit of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.

On conjugacy separability of graph products of groups

We show that the class of $\mathcal{C}$-hereditarily conjugacy separable groups is closed under taking arbitrary graph products whenever the class $\mathcal{C}$ is an extension closed variety of finite groups. As a consequence we show that the class of $\mathcal{C}$-conjugacy separable groups is closed under taking arbitrary graph products. In particular, we show that right angled Coxeter groups are hereditarily conjugacy separable and 2-hereditarily conjugacy separable, and we show that infinitely generated right angled Artin groups are hereditarily conjugacy separable and $p$-hereditarily co…

Some hereditarily just infinite subgroups of the Nottingham Group

This work examines the commutator structure of some closed subgroups of the wild group of automorphisms of a local field with perfect residue field, a group we call $\Cal J.$ In particular, we establish a new approach to evaluating commutators in $\Cal J$ and using this method investigate the normal subgroup structure of some classes of index subgroups of $\Cal J$ as introduced by Klopsch. We provide new proofs of Fesenko\rq s results that lead to a proof that the torsion free group $T =\{t+\sum_{k\geq 1} a_kt^{qk+1}: a_k \in \Bbb F_p\}$ is hereditarily just infinite, and by extending his work…

$\boldsymbol{\textit{C}}$-遗传共轭可分群与 wreath 积

$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products

该研究给出限制 wreath 积为 $\boldsymbol{\textit{C}}$-遗传共轭可分群的充要条件，证明 Grigorchuk 群是 2-遗传共轭可分群，构造了无穷多类非多循环遗传共轭可分群。

Embedding properties of hereditarily just infinite profinite wreath products

We study infinitely iterated wreath products of finite permutation groups with respect to product actions. In particular, we prove that, for every non-empty class of finite simple groups $\mathcal{X}$, there exists a finitely generated hereditarily just infinite profinite group $W$ with composition factors in $\mathcal{X}$ such that any countably based profinite group with composition factors in $\mathcal{X}$ can be embedded into $W$. Additionally we investigate when infinitely iterated wreath products of finite simple groups with respect to product actions are co-Hopfian or non-co-Hopfian.

Hereditarily separable groups and monochromatic uniformization

We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality aleph_1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that Z is a p.i.d. with infinitely many primes is essential for this result.

具有正第一 ℓ²-Betti 数的遗传 just-infinite 挠群

Hereditarily just-infinite torsion groups with positive first $\ell^2$-Betti number

本文提出一种基于 profinite 视角的新方法构造有限生成、剩余有限的无限挠群，首次给出具有正第一 ℓ²-Betti 数的遗传 just-infinite 群例子，并证明其正规子群增长为多项式，回答了 Barnea 和 Schlage-Puchta 的问题。

关于Thompson群$F$及其他遗传可分群的极限群

Limit groups with respect to Thompson's group $F$ and other hereditarily separating groups

本文证明Thompson群$F$上的极限群不能是$F$关于有限生成子群的HNN-扩张，但给出了中心化HNN-扩张的例子，并研究了相对极限群。

On hereditarily self-similar $p$-adic analytic pro-$p$ groups

A non-trivial finitely generated pro-$p$ group $G$ is said to be strongly hereditarily self-similar of index $p$ if every non-trivial finitely generated closed subgroup of $G$ admits a faithful self-similar action on a $p$-ary tree. We classify the solvable torsion-free $p$-adic analytic pro-$p$ groups of dimension less than $p$ that are strongly hereditarily self-similar of index $p$. Moreover, we show that a solvable torsion-free $p$-adic analytic pro-$p$ group of dimension less than $p$ is strongly hereditarily self-similar of index $p$ if and only if it is isomorphic to the maximal pro-$p$…

遗传性与超广义 co-Bassian 阿贝尔群

Hereditarily and Super Generalized co-Bassian Abelian Groups

本文通过给出充分必要条件，完整刻画了具有遗传性质或超性质的 co-Bassian 及广义 co-Bassian 阿贝尔群，深化了对这两类群结构的理解。

Arnautov's problems on semitopological isomorphisms

Semitopological isomorphisms of topological groups were introduced by Arnautov, who posed several questions related to compositions of semitopological isomorphisms and the groups G (we call them Arnautov groups) such that for every group topology T on G every semitopological isomorphism with domain (G,T) is necessarily open (i.e., a topological isomorphism). We propose a different approach to these problems by introducing appropriate new notions, necessary for a deeper understanding of Arnautov groups. This allows us to find some partial answers and many examples. In particular, we discuss the…

关于迭代 wreath 产物在积作用下的子群增长

On subgroup growth of iterated wreath products in product action

本文研究了通过迭代wreath产物在积作用下得到的hereditarily just infinite profinite群的子群增长类型，证明了存在任意子群增长类型的群。

混合恒等式、遗传分离作用与振荡

Mixed identities, hereditarily separated actions and oscillation

本文针对拓扑群作用，提出带参数词的一般条件以保证不等式有解，并应用于Thompson群F和分支群，推广了第二作者博士论文中的结果。

On w-maximal groups

Let $w = w(x_1,..., x_n)$ be a word, i.e. an element of the free group $F =<x_1,...,x_n>$ on $n$ generators $x_1,..., x_n$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{w (g_1,...,g_n)^{\pm 1} | g_i \in G, 1\leq i\leq n \}$ of all $w$-values in $G$. We say that a (finite) group $G$ is $w$-maximal if $|G:w(G)|> |H:w(H)|$ for all proper subgroups $H$ of $G$ and that $G$ is hereditarily $w$-maximal if every subgroup of $G$ is $w$-maximal. In this text we study $w$-maximal and hereditarily $w$-maximal (finite) groups.

Interpreting the arithmetic in Thompson's group F

We prove that the elementary theory of Thompson's group $F$ is hereditarily undecidable.

On zero-dimensionality and the connected component of locally pseudocompact groups

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is dense in the component of the completion of G, and G/G_0 is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.

On subgroup conjugacy separability of hyperbolic QVH-groups

A group $G$ is called subgroup conjugacy separable (abbreviated as SCS) if any two finitely generated and non-conjugate subgroups of $G$ remain non-conjugate in some finite quotient of $G$. An into-conjugacy version of SCS is abbreviated by SICS. We prove that if $G$ is a hyperbolic group, $H_1$ is a quasiconvex subgroup of $G$, and $H_2$ is a subgroup of $G$ which is elementwise conjugate into $H_1$, then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. As corollary, we deduce that fundamental groups of closed hyperbolic 3-manifolds and torsion-free small cancellat…

所有真商群为殆幂零的群的结构

The structure of groups with all proper quotients virtually nilpotent

本文研究所有真商群均为殆幂零的群（JNN$_c$F 群），给出其有限生成 profinite 情形的刻画、结构定理及遗传性判定，并构造遗传 JNN$_c$F 群。

One-relator groups with torsion are conjugacy separable

We prove that one-relator groups with torsion are hereditarily conjugacy separable. Our argument is based on a combination of recent results of Dani Wise and the first author. As a corollary we obtain that any quasiconvex subgroup of a one-relator group with torsion is also conjugacy separable.

Ideal boundary of 7-systolic complexes and groups

We prove that ideal boundary of a 7-systolic group is strongly hereditarily aspherical. For some class of 7-systolic groups we show their boundaries are connected and without local cut points, thus getting some results concerning splittings of those groups.

关于Skiba问题的Galt-Tyutyanov猜想的一个证明

A proof of the Galt--Tyutyanov conjecture on Skiba's question

本文通过抛物子群下降方法与有限单群分解分类，证明了Galt-Tyutyanov猜想，给出有限群为单群的新判据，部分回答Skiba问题，此前仅对部分单群验证过该猜想。

Dense-separable groups and its applications in $d$-independence

A topological space is called {\it dense-separable} if each dense subset of its is separable. Therefore, each dense-separable space is separable. We establish some basic properties of dense-separable topological groups. We prove that each separable space with a countable tightness is dense-separable, and give a dense-separable topological group which is not hereditarily separable. We also prove that, for a Hausdorff locally compact group , it is locally dense-separable iff it is metrizable. Moreover, we study dense-subgroup-separable topological groups. We prove that, for each compact torsion …

Normal subgroups of SimpHAtic groups

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups. The proof relies on studying homotopy types at infinity of groups in question. We also show that nonuniform lattices in SimpHAtic complexes (and in more g…

On generation problems in generalised Wilson type groups

We study a family of hereditarily just infinite profinite groups obtained by iterated wreath products introduced by J. Wilson in 2010. We find explicit generators for this family in a number of cases using combinatorial methods. We then discuss determination of the lower rank for this family. Finally we examine whether the groups in this family admit a finite profinite presentation.

Conjugacy in normal subgroups of hyperbolic groups

Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the hyperbolic group from F. Haglund's and D. Wise's version of Rips's construction is hereditarily conjugacy separable. We then use this construction to produce first examples of finitely generated and finitely presented conjugacy separable groups that contain non-(conjugacy separable) subgroups of finite index.

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…

Characterizing sequences for precompact group topologies

A precompact group topology $τ$ on an abelian group $G$ is called {\em single sequence characterized} (for short, {\em ss-characterized}) if there is a sequence $\mathbf{u}= (u_n)$ in $G$ such that $τ$ is the finest precompact group topology on $G$ making $\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group $(G,τ)$ is $ss$-characterized iff it is countable. For every metrizable precompact group topology $τ$ on a countably infinite abelian group $G$ there exists a group topology $η$ such that $η$ is strictly finer than $τ$ and the groups $(G,τ)$ and $(G,η…

On finitely generated normal subgroups of right-angled Artin groups and graph products of groups

A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we …

The Markov-Zariski topology of an abelian group

According to Markov, a subset of an abelian group G of the form {x in G: nx=a}, for some integer n and some element a of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (=totally bounded) Hausdorff group topology on G. The family of all algebraic subsets of an abelian group G forms the family of closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or verbal, topology of G. We investigate the properties of this …

Moufang sets of finite Morley rank of odd type

We show that for a wide class of groups of finite Morley rank the presence of a split $BN$-pair of Tits rank $1$ forces the group to be of the form $\operatorname{PSL}_2$ and the $BN$-pair to be standard. Our approach is via the theory of Moufang sets. Specifically, we investigate infinite and so-called hereditarily proper Moufang sets of finite Morley rank in the case where the little projective group has no infinite elementary abelian $2$-subgroups and show that all such Moufang sets are standard (and thus associated to $\operatorname{PSL}_2(F)$ for $F$ an algebraically closed field of chara…

群的极限簇的显式构造

An explicit construction of a limit variety of groups

本文构造了首个生成元与基均显式给出的群的极限簇，该簇无有限恒等式基但真子簇均有有限基，相关结果丰富了群簇理论的研究。

关于直有限 Abel 群的一些遗传类与超类

On Some Hereditary and Super Classes of Directly Finite Abelian Groups

本文研究直有限 Abel 群相关类的遗传与超性质，证明相对 Hopf 群具备相应性质当且仅当它是扩张 Bassian 的，并给出更一般背景下的充要条件。