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Transactions of the American Mathematical Society

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



Semistability at $ \infty$, $ \infty$-ended groups and group cohomology

Author: Michael L. Mihalik
Journal: Trans. Amer. Math. Soc. 303 (1987), 479-485
MSC: Primary 20E06; Secondary 20J05, 57M10
MathSciNet review: 902779
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Abstract: A finitely presented group $ G$, is semistable at $ \infty $ if for some (equivalently any) finite complex $ X$, with $ {\pi _1}(X) = G$, any two proper maps $ r,\,s:[0,\,\infty ) \to \tilde X$ ($ \equiv $ the universal cover of $ X$) that determine the same end of $ \tilde X$ are properly homotopic in $ \tilde X$.

If $ G$ is semistable at $ \infty $, then $ {H^2}(G;\,ZG)$ is free abelian. 0- and $ 2$-ended groups are all semistable at $ \infty $.

Theorem. If $ G = A{{\ast}_C}B$ where $ C$ is finite and $ A$ and $ B$ are finitely presented, semistable at $ \infty $ groups, then $ G$ is semistable at $ \infty $.

Theorem. If $ \alpha :C \to D$ is an isomorphism between finite subgroups of the finitely presented semistable at $ \infty $ group $ H$, then the resulting $ HNN$ extension is semistable at $ \infty $.

Combining these results with the accessibility theorem of M. Dunwoody gives

Theorem. If all finitely presented $ 1$-ended groups are semistable at $ \infty $, then all finitely presented groups are semistable at $ \infty $.

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Article copyright: © Copyright 1987 American Mathematical Society