Corrigendum to “Height graded relative hyperbolicity and quasiconvexity”

François Dahmani,Mahan Mj

doi:10.5802/jep.97

Abstract

We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex cocompactness in mapping class groups and $Out(F_n)$. Corrigendum: there is an unfortunate mistake in the statement and the proof of Proposition 5.1. This affects one direction of the implications of the main theorem. A correction is given, that states that given a quasi-convex subgroup of a hyperbolic (or relatively hyperbolic) group, the graded relative hyperbolic structure holds with respect to saturations of I-fold intersections, that are stabilizers of limit sets of I-fold intersections.

Highlights

Résumé (Correction à « Hauteur, hyperbolicité relative graduée, et quasiconvexité ») Une malencontreuse erreur entache la preuve, et l’énoncé, de la Proposition 5.1 de l’article mentionné en titre
In the case As and Bs are unbounded, we first observe that in that metric, ΛAs ∩ ΛgBsg−1 = Λ(As ∩ gBsg−1); for proving this we distinguish whether G is a relatively hyperbolic group or not
– If (G, {H}, d) has graded geometric relative hyperbolicity and the action of H on the curve complex is uniformly proper, H is convex cocompact in Mod(S). (4) Let G be Out(Fn) and d the metric obtained by electrifying the subgroups corresponding to subgroups that stabilize proper free factors so that (G, d) is quasi-isometric to the free factor complex Fn

Summary

Main modifications and corrections

The original version of Proposition 5.1 asserted that (G, {H}, d) had the geometric graded relative hyperbolicity property Any three-fold intersection is trivial, the metric d3 is the word metric on F over {a, b} This metric is not relatively hyperbolic with respect to the two-fold intersections (i.e. the conjugates of H) as was incorrectly promised by Proposition 5.1, because H has two cosets that remain at bounded distance of one another. It is relatively hyperbolic with respect to the conjugates of the stabilizer of the limit set of H (which is a , and is its own normalizer). This last statement can be generalized to the setting of Proposition 5.1, and is the purpose of this corrigendum

Definition of saturation

Uniform qi intersection property passes to saturations

Conclusion