Tree-graded spaces and asymptotic cones of groups

TL;DR

This paper introduces tree-graded spaces to analyze the large-scale geometry of groups, establishing invariance properties, characterizing relatively hyperbolic groups via asymptotic cones, and providing new examples of groups with diverse asymptotic cones.

Contribution

It develops the concept of tree-graded spaces and applies it to characterize relatively hyperbolic groups and their asymptotic cones, including the first example with a continuum of non-equivalent cones.

Findings

Quasi-isometry invariance of certain relatively hyperbolic groups

Characterization of relatively hyperbolic groups through asymptotic cones

Construction of a finitely generated group with a continuum of non-$$-equivalent asymptotic cones

Abstract

We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of finitely generated group with a continuum of non-$\pi_1$-equivalent asymptotic cones. Note that by a result of Kramer, Shelah, Tent and Thomas, continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true.