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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.

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