Tree-graded asymptotic cones

TL;DR

This paper investigates the structure of asymptotic cones of certain groups, revealing their dependence on the bilipschitz types of pieces in tree-graded structures and providing explicit descriptions.

Contribution

It establishes that asymptotic cones of relatively hyperbolic groups depend only on the bilipschitz types of their pieces, and describes these cones explicitly.

Findings

Asymptotic cones depend only on the bilipschitz types of pieces.

Many relatively hyperbolic groups have asymptotic cones independent of scaling.

Explicit descriptions of asymptotic cones are provided.

Abstract

We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of relatively hyperbolic groups (resp. asymptotic cones of groups containing a cut-point) only depend on the bilipschitz equivalence types of the pieces in the standard (resp. minimal) tree-graded structure. In particular, the asymptotic cones of many relatively hyperbolic groups do not depend on the scaling factor. We also describe the asymptotic cones as above "explicitly". Part of these results were obtained independently and simultaneously by D. Osin and M. Sapir.