A Manual for Ends, Semistability and Simple Connectivity at Infinity for Groups and Spaces

TL;DR

This paper provides a comprehensive, updated overview of the current understanding of ends, semistability, and simple connectivity at infinity for groups and spaces, including new results and historical context.

Contribution

It compiles and discusses the latest results on these topological properties, introduces new findings, and offers a detailed index to aid researchers in the field.

Findings

Updated results on semistability and simple connectivity at infinity

New insights into the second cohomology of groups

Inclusion of mapping class groups and new indices

Abstract

This $2^{nd}$-edition article is intended to be an up-to-date archive of the current state of the questions: Which finitely generated groups $G$: have semistable fundamental group at infinity; are simply connected at infinity; are such that $H^2(G,\mathbb ZG)$ is free abelian or trivial. The idea is not to reprove these results, but to provide a historical record of the progress on these questions and provide a list of the most general results. We also prove or cite all of the results that make up the basic theory. The first Chapter is devoted to ends of groups and spaces, and the second to semistability at infinity, simple connectivity at infinity and second cohomology of groups. Definitions, basic facts and lists of general results are given in each Chapter. A number of results proven here are new and a number of authors have contributed results. We end with an Index for simply connected at infinity groups and an Index of Groups and Subgroups which is intended to help a reader quickly locate results about certain types of groups/subgroups. The main updates from the first edition is section 2.4.5 on mapping class groups and the addition of the simply connected at infinity index.