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Commensurated subgroups, semistability and simple connectivity at infinity

Gregory R Conner and Michael L Mihalik

Algebraic & Geometric Topology 14 (2014) 3509–3532

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 one-ended and semistable at . Furthermore, if Q and G are finitely presented and either Q is one-ended or the pair (G,Q) has one 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 [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 G 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 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 .

commensurator, semistability, simply connected at infinity
Mathematical Subject Classification 2010
Primary: 20F69
Secondary: 20F65
Received: 16 September 2013
Revised: 3 March 2014
Accepted: 19 March 2014
Published: 15 January 2015
Gregory R Conner
Math Department
Brigham Young University
275 TMCB
Provo, UT 84602
Michael L Mihalik
Mathematics Department
Vanderbilt University
Nashville, TN 37240