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

TL;DR

This article compiles a comprehensive archive of ends, semistability at infinity, and simple connectivity at infinity for groups and spaces, integrating foundational definitions with extensive results across broad group classes. It develops the framework of ends, pro-groups, and Cayley 2-complexes, then surveys how semistability and simple connectivity at infinity manifest in finite, 2-ended, and particularly 1-ended groups, including mapping class groups, Out($F_n$), Artin/Coxeter groups, solvable and metanilpotent groups, and relatively hyperbolic groups. Key methodological pivots include graph-of-groups decompositions, ascending HNN extensions, pro-mono and coaxial actions, and boundary-structure arguments for hyperbolic/CAT(0) groups, illustrating when a group is semistable at infinity or simply connected at infinity. The work highlights important results such as Stallings/Dunwoody accessibility for ends, Touikan's end-theory for graphs of groups, and the transfer of end-behavior to algebraic properties like $H^2(G, obreak Z G)$, with explicit treatments of mapping class groups and outer automorphism groups. Overall, the paper functions as a foundational reference, consolidating definitions, equivalences, and a wide spectrum of semistability and connectivity outcomes—crucial for understanding the large-scale topology of groups and their spaces. It also points to active areas (e.g., CAT(0) group boundaries and relative hyperbolicity) where semistability at infinity remains nuanced and central to ongoing research.

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.

Figures (15)

• Figure 1: Separating elements of $H$
• Figure 2: Separating elements of $H$
• Figure 3: $r$ is properly homotopic to $s$
• Figure 4: Stacking Homotopies
• Figure 5: The case $v=r(x_1)$

Theorems & Definitions (283)

• Theorem 1.1.1
• Remark 1.1.2
• Theorem 1.1.3
• Theorem 1.1.4
• Theorem 1.1.5
• Theorem 1.1.6: Proposition 13.4.1, G
• Theorem 1.1.7
• Example 1.1.8
• Theorem 1.1.9
• Lemma 1.1.10