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

DOI:
https://doi.org/10.1090/S0002-9947-1990-0961627-X

MathSciNet review:
961627

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Abstract: A CW-complex is simply connected at infinity if for each compact in there exists a compact in such that loops in are homotopically trivial in . Let be a finitely presented group and a finite CW-complex with fundamental group . is said to be simply connected at infinity if the universal cover of is simply connected at infinity. B. Jackson and C. M. Houghton have independently shown that if and a normal subgroup are infinite finitely presented groups with infinite and either or -ended, then is simply connected at infinity. In the case where is -ended, we exhibit a class of groups showing that the "finitely presented" hypothesis on cannot be reduced to "finitely generated." We address the question: if is normal in and is normal in and these are infinite groups with and finitely presented and either or is -ended, is simply connected at infinity? In the case that is -ended, the answer is shown to be yes. In the case that is -ended, we exhibit a class of such groups that are not simply connected at infinity.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-0961627-X

Keywords:
Universal cover,
CW-complex,
group representations,
homotopy

Article copyright:
© Copyright 1990
American Mathematical Society