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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Semistability at the end of a group extension


Author: Michael L. Mihalik
Journal: Trans. Amer. Math. Soc. 277 (1983), 307-321
MSC: Primary 57M05; Secondary 20F32, 57M10
DOI: https://doi.org/10.1090/S0002-9947-1983-0690054-4
MathSciNet review: 690054
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Abstract: A $ 1$-ended $ {\text{CW}}$-complex, $ Q$, is semistable at $ \infty $ if all proper maps $ r:\ [0,\infty) \to Q$ are properly homotopic. If $ {X_1}$ and $ {X_2}$ are finite $ {\text{CW}}$-complexes with isomorphic fundamental groups, then the universal cover of $ {X_1}$ is semistable at $ \infty $ if and only if the universal cover of $ {X_2}$ is semistable at $ \infty $. Hence, the notion of a finitely presented group being semistable at $ \infty $ is well defined. We prove

Main Theorem. Let $ 1 \to H \to G \to K \to 1$ be a short exact sequence of finitely generated infinite groups. If $ G$ is finitely presented, then $ G$ is semistable at $ \infty $.

Theorem. If $ A$ and $ B$ are locally compact, connected noncompact $ CW$-complexes, then $ A \times B$ is semistable at $ \infty $.

Theorem. $ \langle\;x,y:x{y^b}{x^{ - 1}} = {y^c};b\; and \; c \; nonzero\; integers\; \rangle $ is semistable at $ \infty $.

The proofs are geometrical in nature and the main tool is covering space theory.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0690054-4
Article copyright: © Copyright 1983 American Mathematical Society