The Schur multiplier, profinite completions and decidability

Direct factors of profinite completions and decidability

We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$.

Profinite completion of Grigorchuk's group is not finitely presented

In this paper we prove that the profinite completion $\mathcal{\hat G}$ of the Grigorchuk group $\mathcal{G}$ is not finitely presented as a profinite group. We obtain this result by showing that $H^2(\mathcal{\hat G},\field{F}_2)$ is infinite dimensional. Also several results are proven about the finite quotients $\mathcal{G}/ St_{\mathcal{G}}(n)$ including minimal presentations and Schur Multipliers.

有限半群和自动机的克罗恩-罗兹复杂度c=1的可判定性

Decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata

本文研究了有限半群和自动机的克罗恩-罗兹复杂度c=1的可判定性，通过证明亨克尔、罗兹和斯坦伯格2012年的下界结果在使用 profinite 方法和麦卡蒙德1991年和2001年的结果后是精确的，解决了计算复杂度的算法问题。

所有有限半群和自动机的Krohn-Rhodes复杂度的可判定性

Decidability of Krohn-Rhodes complexity for all finite semigroups and automata

本文证明了对于任意大于等于0的k，有限半群是否具有Krohn-Rhodes复杂度k是可判定的，通过改进2012年论文中的下界并证明其对任意复杂度均精确，解决了50多年的开放问题。

Remarks on Proficient groups

If a finite group G has a presentation with d generators and r relations, it is well-known that r - d is at least the rank of the Schur multiplier of G; a presentation is called efficient if equality holds. There is an analogous definition for proficient profinite presentations. We show that many perfect groups have proficient presentations. Moreover, we prove that infinitely many alternating groups, symmetric groups and their double covers have proficient presentations

舒尔定理的一种动力学方法

A dynamical approach to Schur's Theorem

该研究为局部紧群建立舒尔定理的新动力学解释，证明连通Z-群的导出子群闭包具有限拓扑熵，还在海森堡群框架下得到p-秩新公式，完善了舒尔定理的相关研究。