The generalisation of some invariants of modules to groups with operators

Baumslag–Solitar 群及其推广的上同调可分性

Cohomological Separability of Baumslag--Solitar groups and Their Generalisations

本文在广义 Baumslag–Solitar 群类中给出上同调可分性的等价刻画，并对 Baumslag–Solitar 群得到上同调可分性与有限完备化上同调维数的三分法，从而构造出具有和不具有可分上同调的非剩余有限单关系群例子。

Some generalisations of Schur's and Baer's theorem and their connection with homological algebra

Schur's Theorem and its generalisation, Baer's Theorem, are distinguished results in group theory, connecting the upper central quotients with the lower central series. The aim of this paper is to generalise these results in two different directions, using novel methods related with the non-abelian tensor product. In particular, we prove a version of Schur-Baer Theorem for finitely generated groups. Then, we apply these newly obtained results to describe the $k$-nilpotent multiplier, for $k\geq 2$, and other invariants of groups.

Structure trees, networks and almost invariant sets

A self-contained account of the theory of structure trees for edge cuts in networks is given. Applications include a generalisation of the Max-Flow Min-Cut Theorem to infinite networks and a short proof of a conjecture of Kropholler. This gives a relative version of Stallings' Theorem on the structure of groups with more than one end. A generalisation of the Almost Stability Theorem is also obtained, which provides information about the structure of the Sageev cubing.

On induced modules of inertial-invariant support $τ$-tilting modules over blocks of finite groups

In this article, we prove that induced modules of support $τ$-tilting modules over blocks of finite groups satisfying inertial-invariant condition are also support $τ$-tilting modules.

The Benson -- Symonds Invariant for Ordinary and Signed Permutation Modules

The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $γ_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.

不变量模环的相对不变式、理想类和拟典范模

Relative Invariants, Ideal Classes and Quasi-Canonical Modules of Modular Rings of Invariants

本文描述了阶乘Gorenstein整环上有限群作用的模不变环$R$的“拟正则模”.由此，我们得到了一个一般的“准Gorenstein准则”的某些1-上圈。这推广了A. Braun关于多项式环上线性群作用的结果，推广了Watanabe关于非模不变环的一个经典结果.我们使用一个显式分类的所有自反秩一$R$ -模块，这是在类组的$R$，或在$R$ -半不变量。这一结果隐含在Nakajima的一篇论文（\cite{Nakajima:rel_inv}）中。