Jan Saxl, Saharon Shelah, Simon Thomas
We classify those sequences $\langle S_{n} \mid n \in \mathbb{N} \rangle$ of finite simple nonabelian groups such that the full product $\prod_{n} S_{n}$ has property (FA).
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We characterise the group property of being with infinite conjugacy classes for wreath products of groups
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.
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Recently, the authors gave some conditions under which a direct product of finitely many groups is $\mathcal{V}-$capable if and only if each of its factors is $\mathcal{V}-$capable for some varieties $\mathcal{V}$. In this paper, we extend this fact to any infinite direct product of groups. Moreover, we conclude some results for $\mathcal{V}-$capability of direct products of infinitely many groups in varieties of abelian, nilpotent and polynilpotent groups.
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.
We study a class of quasimorphisms of the free group that can be expressed as infinite sums of Brooks quasimorphisms with some nice properties. We then review Heuer's framework of decompositions developed in arXiv:1710.03193, and put these quasimorphisms in that context. This allows us to prove triviality of a number of cup products in the bounded cohomology of the free group with trivial real coefficients. Finally, we introduce free products of decompositions and relate them to Rolli's quasimorphisms of free products. A technical result concerning these, which remains an open question, implie…
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If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.
Groups of almost upper triangular infinite matrices with entries indexed by integers are studied. It is shown that, when the matrices are over a finite field, these groups admit a nondiscrete totally disconnected, locally compact group topology and are topologically simple.
In this paper we show that there exists an uncountable family of finitely generated simple groups with the same positive theory as any non-abelian free group. In particular, these simple groups have infinite $w$-verbal width for all non-trivial words $w$.
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Groups with infinite linearly ordered products
本文提出无限线性有序乘积的形式体系，用于定义超序列等幂级数群中的无限复合，并证明此类群可表示为有序阿贝尔群的半直积。
Infinite free pro-p products as absolute Galois groups
本文推广梅利尼科夫准则，建立自由 profinite 与自由 pro-p 乘积实现为绝对伽罗瓦群的联系。
We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on nonabelian simple groups.
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Infinite graph product of groups II: Analytic properties
研究具有双曲定义图的有限群的图积的解析性质，通过扩展图或交叉图的Bowditch紧化上的动力学，给出了收敛群的新类，并确定了该收敛作用为几何有限的充要条件，同时提供了适当近端群、相对双精确群和具有强固体群von Neumann代数的群的新类。
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本文引入图群的扩展图并研究其几何性质，证明扩展图与准中位图的交叉图同构，并应用该图证明图 wreath 产品的相对超几何性，提供新的相对超几何群构造方法。
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