# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Semistability at $\infty$, $\infty$-ended groups and group cohomologyHTML articles powered by AMS MathViewer

by Michael L. Mihalik
Trans. Amer. Math. Soc. 303 (1987), 479-485 Request permission

## 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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