发表机构
Institute for Mathematics, Physics and Mechanics; Faculty of Mathematics and Physics, University of Ljubljana（数学、物理与力学研究所; 卢布尔雅那大学数学与物理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出了与群作用相关的新不变量tc(d)，发展其一般理论并计算了映射环面基本群等例子中的取值，将其与上同调维数等概念关联。
AI 中文摘要
我们引入了一个与群G在（可能非交换的）群K上的作用相关联的新不变量。该不变量基于半直积K⋊G的G参数化LS范畴，同时也依赖于对导子d∶G→K的选择，因此记为tc(d)，其取值为介于cd(K)和cd(K)+cd(G)之间的整数（其中cd(G)表示群的上同调维数）。我们发展了一般理论，随后计算了多个重要例子中该不变量的取值，包括映射环面的基本群、纯辫群的基本群以及自由群的迭代半直积。为实现这一点，我们将tc(d)与其因子的上同调维数、G在K⋊G中的包含映射的截面范畴，以及与导子d相关联的挠子的性质联系起来。
英文摘要
We introduce a new invariant associated to an action of a group $G$ on a (possibly non-commutative) group $K$. The invariant is based on the $G$-parametrized LS-category of the semi-direct product $K\rtimes G$ but it also depends on a choice of a derivation $d\colon G\to K$. The invariant is thus denoted $\operatorname{tc}(d)$ and it takes integer values between $\operatorname{cd}(K)$ and $\operatorname{cd}(K)+\operatorname{cd}(G)$ (where $\operatorname{cd}(G)$ stands for the cohomological dimension of a group). We develop a general theory and then compute its values for a number of important examples, including fundamental groups of mapping tori, of pure braid groups and of iterated semi-direct products of free groups. To achieve this, we relate $\operatorname{tc}(d)$ to the cohomological dimension of its factors, to the sectional category of the inclusion of $G$ in $K\rtimes G$, and to the properties of torsors that are associated to the derivation $d$.
Comments19 pages
