Commensurated Subgroups, Semistability and Simple Connectivity at Infinity

A subgroup Q of a group G is commensurated if the commensurator of Q in G is the entire group G. Our main result is that a finitely generated group G containing an infinite, finitely generated, commensurated subgroup H, of infinite index in G is 1-ended and semistable at ∞. If additionally, Q and G are finitely presented and either Q is 1-ended or the pair (G,Q) has 1 filtered end, then G is simply connected at∞. A normal subgroup of a group is commensurated, so this result is a generalization of M. Mihalik’s result in [18] and of B. Jackson’s result in [13]. As a corollary, we give an alternate proof of V. M. Lew’s theorem that a finitely generated group G containing an infinite, finitely generated, subnormal subgroup of infinite index is semistable at ∞. So, many previously known semistability and simple connectivity at ∞ results for group extensions follow from the results in this paper. If φ : H → H is a monomorphism of a finitely generated group and φ(H) has finite index in H, then H is commensurated in the corresponding ascending HNN extension, which in turn is semistable at ∞.