Abstract
A subgroup of a group is commensurated if the commensurator of in is the entire group . Our main result is that a finitely generated group containing an infinite, finitely generated, commensurated subgroup of infinite index in is one-ended and semistable at . Furthermore, if and are finitely presented and either is one-ended or the pair has one filtered end, then is simply connected at . A normal subgroup of a group is commensurated, so this result is a generalization of M Mihalik’s result [Trans. Amer. Math. Soc. 277 (1983) 307–321] and of B Jackson’s result [Topology 21 (1982) 71–81]. As a corollary, we give an alternate proof of V M Lew’s theorem that a finitely generated group containing an infinite, finitely generated, subnormal subgroup of infinite index is semistable at . So several previously known semistability and simple connectivity at results for group extensions follow from the results in this paper. If is a monomorphism of a finitely generated group and has finite index in , then is commensurated in the corresponding ascending HNN extension, which in turn is semistable at .
Citation
Gregory R Conner. Michael L Mihalik. "Commensurated subgroups, semistability and simple connectivity at infinity." Algebr. Geom. Topol. 14 (6) 3509 - 3532, 2014. https://doi.org/10.2140/agt.2014.14.3509
Information