Commensurated subgroups, semistability and simple connectivity at infinity

Conner, Gregory R¹ ; Mihalik, Michael L²

¹Math Department, Brigham Young University, 275 TMCB, Provo, UT 84602, USA
²Mathematics Department, Vanderbilt University, Nashville, TN 37240, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3509-3532

Résumé

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 ∞.

Détail

DOI : 10.2140/agt.2014.14.3509

Classification : 20F69, 20F65
Keywords: commensurator, semistability, simply connected at infinity

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