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Near Ascending HNN-Extensions and a Combination Result for Semistability at Infinity

Semistability at infinity is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at infinity for a 1-ended group. It is an open problem whether or not all finitely presented groups have semistable fundamental group at infinity. While many classes of groups are known to contain only semistable at infinity groups, there are only a few combination results for such groups. Our main theorem is such a result. Main Theorem. Suppose $G$ is the fundamental group of a connected reduced graph of groups, where each edge group is infinite and finitely generated, and each vertex group is finitely presented and either 1-ended and semistable at infinity or has an edge group of finite index. Then $G$ is 1-ended and semistable at infinity. An important part of the proof of this result is the semistability part of the following: Theorem. Suppose $H_0$ is an infinite finitely presented group, $H_1$ is a subgroup of finite index in $H_0$, $ϕ:H_1\to H_0$ is a monomorphism and $G=H_0\ast_ϕ$ is the resulting HNN extension. Then $G$ is 1-ended and semistable at infinity. If additionally, $H_0$ is 1-ended, then $G$ is simply connected at infinity.