Quasi‐isometries of relatively hyperbolic groups with an elementary hierarchy

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 locally extended residually finite (LERF) and virtually toral relatively hyperbolic.