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Transactions of the American Mathematical Society

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



Using subnormality to show the simple connectivity at infinity of a finitely presented group

Author: Joseph S. Profio
Journal: Trans. Amer. Math. Soc. 320 (1990), 281-292
MSC: Primary 20F05; Secondary 55Q05, 57M20
MathSciNet review: 961627
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Abstract: A CW-complex $ X$ is simply connected at infinity if for each compact $ C$ in $ X$ there exists a compact $ D$ in $ X$ such that loops in $ X - D$ are homotopically trivial in $ X - C$. Let $ G$ be a finitely presented group and $ X$ a finite CW-complex with fundamental group $ G$. $ G$ is said to be simply connected at infinity if the universal cover of $ X$ is simply connected at infinity. B. Jackson and C. M. Houghton have independently shown that if $ G$ and a normal subgroup $ H$ are infinite finitely presented groups with $ G/H$ infinite and either $ H$ or $ G/H$ $ 1$-ended, then $ G$ is simply connected at infinity. In the case where $ H$ is $ 1$-ended, we exhibit a class of groups showing that the "finitely presented" hypothesis on $ H$ cannot be reduced to "finitely generated." We address the question: if $ N$ is normal in $ H$ and $ H$ is normal in $ G$ and these are infinite groups with $ N$ and $ G$ finitely presented and either $ N$ or $ G/H$ is $ 1$-ended, is $ G$ simply connected at infinity? In the case that $ N$ is $ 1$-ended, the answer is shown to be yes. In the case that $ G/H$ is $ 1$-ended, we exhibit a class of such groups that are not simply connected at infinity.

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Keywords: Universal cover, CW-complex, group representations, homotopy
Article copyright: © Copyright 1990 American Mathematical Society

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