New Action-Induced Nested Classes of Groups and Jump (Co)homology

TL;DR

This paper introduces a new framework for classifying groups using fixed-point-free actions and extends the concept of virtual cohomological dimension to groups with jump (co)homology, providing new characterizations and classes.

Contribution

It develops a scheme to define nested classes of groups indexed by ordinals and generalizes properties of virtual cohomological dimension to groups with jump (co)homology.

Findings

New classes of groups identified via fixed-point-free actions

Characterizations of well-known classes like virtually polycyclic groups

A subclass of groups shown to have jump (co)homology

Abstract

Using fixed-point-free group actions, we set up a scheme to define nested classes of groups indexed over ordinals. Restricting to cellular actions on CW-complexes, we find new classes as well as new characterizations for some well-known classes, such as virtually polycyclic groups. We generalize properties of the virtual cohomological dimension of a group to groups with jump (co)homology and prove that a core subclass of a new class of groups has jump (co)homology.