**Memoirs of the American Mathematical Society**

1992;
86 pp;
Softcover

MSC: Primary 20;
Secondary 57

Print ISBN: 978-0-8218-2531-0

Product Code: MEMO/98/471

List Price: $29.00

Individual Member Price: $17.40

**Electronic ISBN: 978-1-4704-0897-8
Product Code: MEMO/98/471.E**

List Price: $29.00

Individual Member Price: $17.40

# Semistability of Amalgamated Products and HNN-Extensions

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*Michael L. Mihalik; Steven T. Tschantz*

In the study of the proper homotopy theory of finitely presented groups, semistability at infinity is an end invariant of central importance. A finitely presented group that is semistable at infinity has a well-defined fundamental group at infinity independent of base ray. If \(G\) is semistable at infinity, then \(G\) has free abelian second cohomology with \({\mathbb Z}G\) coefficients. In this work, the authors show that amalgamated products and HNN-extensions of finitely presented semistable at infinity groups are also semistable at infinity. A major step toward determining whether all finitely presented groups are semistable at infinity, this result easily generalizes to finite graphs of groups. In an early application, this result was used in showing that all one-relator groups are semistable at infinity. The theory of group actions on trees and techniques derived from the proof of Dunwoody's accessibility theorem are key ingredients in this work.

#### Table of Contents

# Table of Contents

## Semistability of Amalgamated Products and HNN-Extensions

- Table of Contents v6 free
- §1. Introduction 18 free
- §2. Geometric preliminaries 411 free
- §3. Outline of the proof 1320
- §4. Dunwoody tracks and relative accessibility 1522
- §5. Basic lemmas 2936
- §6. Technical lemmas 4249
- §7. Proof of the half-space lemma 5663
- §8. Proof of theorem 3.3 7178
- §9. Conclusion 8491
- References 8592

#### Readership

Mathematicians interested in geometric group theory, shape theory, or cohomology of groups.