Quasi-isometries of relatively hyperbolic groups with an elementary hierarchy

TL;DR

This paper demonstrates that a class of relatively hyperbolic groups with a hierarchy similar to limit groups is closed under quasi-isometry and shares key properties like being LERF and virtually toral hyperbolic.

Contribution

It introduces a hierarchy-based class of relatively hyperbolic groups and proves their invariance under quasi-isometry, extending properties known for limit groups.

Findings

Groups quasi-isometric to limit groups are LERF.

Such groups are virtually toral relatively hyperbolic.

The class is closed under quasi-isometry.

Abstract

Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. In this paper, a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups, is shown to be closed under quasi-isometry. Additionally, these groups share some of the properties of limit groups. In particular, groups quasi-isometric to limit groups are shown to be LERF and virtually toral relatively hyperbolic.