Height, Graded Relative Hyperbolicity and Quasiconvexity {\it (with Corrigendum)}

TL;DR

This paper introduces new concepts of geometric height and graded relative hyperbolicity to characterize various forms of quasiconvexity across hyperbolic, relatively hyperbolic, and mapping class groups, with a correction to a key proposition.

Contribution

It develops the notions of geometric height and graded relative hyperbolicity, providing a unified framework to understand quasiconvexity and convex cocompactness in diverse group settings.

Findings

Characterizes quasiconvexity in hyperbolic groups

Defines graded relative hyperbolicity for various groups

Provides a corrected statement for key structural implications

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.