Limits over orbit categories of locally finite groups

TL;DR

This work corrects an error in BLO3 and develops a comprehensive theory of higher limits $\Lambda^*(G;M)$ for infinite locally finite groups via orbit categories. It shows how many computations reduce to limits over finite subgroups through two spectral sequences and Kan-extension methods, and it extends Lyndon–Hochschild–Serre-style analyses to orbit-category limits. A key vanishing result states that if $C_G(M)$ contains an element of order $p$, then $\Lambda^*(G;M)=0$, and the paper clarifies when $\Lambda^*(G;M)$ coincides with $\Lambda^*_f(G;M)$ in locally finite settings. Through explicit examples, the authors reveal that infinite locally finite groups can exhibit nontrivial higher limits and behavior not present in the finite case, sharpening our understanding of equivariant cohomology in this broader context.

Abstract

We correct an error in the paper [BLO3], and take the opportunity to examine in more detail the derived functors of inverse limits over orbit categories of (infinite) locally finite groups. The main results show how to reduce this in many cases to limits over orbit categories of finite groups, but we also look at generalizations of the Lyndon-Hochschild-Serre spectral sequence for higher limits over orbit categories for an extension of locally finite groups.

Theorems & Definitions (50)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Definition 1.1
• Proposition 1.2
• proof
• Proposition 1.3
• proof
• Lemma 1.4