Elsevier

Topology and its Applications

Volume 72, Issue 3, 5 September 1996, Pages 273-281
Topology and its Applications

Semistability at infinity, simple connectivity at infinity and normal subgroups

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Abstract

In this paper we prove the following two theorems about the behavior of the fundamental group near ∞ for certain group extensions.

Theorem 1. If NAG are groups with A one-ended, finitely presented and of infinite index in G, G finitely presented, and N a non-locally finite normal subgroup of G, then G is semistable at ∞.

This is a technical result that should provide a useful tool in conjunction with semistability results already in the literature.

Theorem 2. If 1 → HGK → 1 is a short exact sequence of infinite finitely generated groups with G finitely presented, K one-ended and H contained in a finitely presented subgroup L of infinite index in G then G is simply connected at ∞.

This is a generalization of a theorem of B. Jackson. Several applications of this result are discussed in the Introduction.

Finally we construct a negative solution to a long standing problem in the form of a group extension 1 → HGK → 1 where H is a one-ended finitely generated group, G is the non-simply connected at ∞ group (ZnZ) × (ZnZ) and K is the ((n − 2)-connected at ∞) group Zn.

Keywords

Fundamental group
Group extension
Proper homotopy

MSC

57M10
20E22
20F32

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