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

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.

💡 Research Summary

The paper “A Manual for Ends, Semistability and Simple Connectivity at Infinity for Groups and Spaces” (second edition, 2026) is a comprehensive survey and reference work that consolidates the current state of knowledge on three interrelated topics in geometric group theory: (1) the theory of ends of groups and spaces, (2) semistability and simple connectivity at infinity, and (3) the structure of the second cohomology group (H^{2}(G,\mathbb ZG)). Rather than reproving known theorems, the author, Michael Mihalik, collects the most general results, updates them where newer work has appeared, and adds several new contributions. The manuscript is organized into two main chapters, each subdivided into many sections that treat both foundational material and specialized classes of groups.

Chapter 1 – Ends of Groups and Spaces
The chapter begins with a rigorous definition of the space of ends (E(X)) for a connected, locally compact, locally connected, Hausdorff space (X). Using cofinal sequences of compact subsets, the ends are defined as the inverse limit of the sets of unbounded components of the complements. The author proves that (E(X)) is a compact, totally disconnected metrizable space; its cardinality equals the number of ends, and it is finite precisely when the space has finitely many ends. Proper maps induce continuous maps on end spaces, and quasi‑isometries preserve the number of ends. Consequently, the number of ends of a finitely generated group—defined via the Cayley graph—is a quasi‑isometry invariant.

The section on ends of groups reviews the classical work of Hopf and Freudenthal, then explains how the Cayley graph provides a geometric model for the group. The author discusses the dichotomy between 0‑ended (finite) groups, 1‑ended groups (most infinite groups of interest), and groups with two or infinitely many ends. Various structural results are collected: normal or commensurated subgroups constrain the number of ends; combination theorems for amalgamated products and HNN‑extensions describe how ends behave under graph‑of‑groups decompositions; specific families such as knot groups, Artin and Coxeter groups, ascending HNN‑extensions, and groups without free subgroups of rank 2 are examined in detail. The chapter culminates with a discussion of half‑spaces associated to splittings, which provides a useful tool for detecting 2‑ended groups.

Chapter 2 – Semistability, Simple Connectivity at Infinity, and (H^{2}(G,\mathbb ZG))
The second chapter shifts focus to the asymptotic topology of spaces and groups. It begins with the definition of semistability at infinity for a space: roughly, the inverse system of fundamental groups of complements of larger and larger compact sets stabilizes up to surjection. The author introduces the fundamental pro‑group (\check\pi_{1}^{\infty}(X)) and the “stable” fundamental group (\pi_{1}^{\infty}(X)), showing that several classical definitions are equivalent. Proper (k)-equivalences are defined, and it is proved that a proper 1‑equivalence induces a bijection of end spaces and an isomorphism of fundamental pro‑groups.

For groups, the chapter defines semistability and simple connectivity at infinity via the Cayley 2‑complex associated to a finite presentation. The “Cayley 2‑complex” construction (Section 2.3.4) provides a concrete combinatorial model for computing the pro‑group. The author proves that for finitely generated groups, semistability at infinity is independent of the chosen finite generating set or presentation. Simple connectivity at infinity (the vanishing of (\pi_{1}^{\infty})) is shown to be a stronger condition, implying semistability but not conversely.

A substantial portion of the chapter is devoted to cataloguing which families of groups satisfy these asymptotic properties. Highlights include:

• Mapping class groups (newly added in this edition) – shown to be semistable and simply connected at infinity, extending earlier work on surface groups.
• Out((F_n)) – also semistable, using actions on outer space.
• GL((n,\mathbb Z)) – semistable for (n\ge3); (n=2) provides a counterexample.
• Word‑hyperbolic and CAT(0) groups – hyperbolic groups are semistable, CAT(0) groups are simply connected at infinity under mild hypotheses.
• Relatively hyperbolic groups – semistability descends from peripheral subgroups.
• Ascending HNN‑extensions – semistability holds when the associated endomorphism is eventually injective.
• Thompson’s group F – a celebrated example of a finitely presented group that fails semistability at infinity.
• Lamplighter groups – another explicit non‑semistable example, illustrating how wreath products can produce pathological behavior.
• Artin and Coxeter groups, Bieri–Stallings groups, solvable and metanilpotent groups, graph products, and many others are treated, often with precise criteria (e.g., presence of a normal infinite cyclic subgroup, existence of a 2‑ended splitting, etc.).

Section 2.5 connects these topological properties to the algebraic question about the second cohomology group (H^{2}(G,\mathbb ZG)). The author explains how the problem can be reduced to the case of 1‑ended groups, and how semistability, 1‑acyclicity at infinity, and the pro‑finite first homology at infinity are equivalent formulations. A catalogue of homological results is provided, summarizing known vanishing theorems and describing when (H^{2}(G,\mathbb ZG)) is free abelian, trivial, or more exotic.

The paper concludes with two extensive indices: one listing all groups known to be simply connected at infinity, and another indexing groups and subgroups together with the relevant results proved or cited in the text. These indices are designed to allow researchers to quickly locate the status of a particular group with respect to ends, semistability, simple connectivity, and second cohomology.

Overall Assessment
Mihalik’s manual serves as an indispensable reference for anyone working in geometric group theory, low‑dimensional topology, or asymptotic group invariants. By gathering definitions, equivalences, and the most up‑to‑date theorems in a single, well‑organized document, the author eliminates the need to chase down dozens of original papers. The inclusion of new results (especially the mapping class group section) and the systematic treatment of counterexamples (Thompson’s F, lamplighter groups) give the work both breadth and depth. The clear exposition of the relationship between ends, semistability, simple connectivity at infinity, and the algebraic invariant (H^{2}(G,\mathbb ZG)) makes the manual a valuable bridge between geometric intuition and algebraic consequences. Researchers can use it both as a teaching resource and as a quick lookup for the current status of any finitely generated group concerning its asymptotic topology.