Universal tree-graded spaces and asymptotic cones

TL;DR

This paper constructs the universal tree-graded space with specified pieces and uses it to analyze the uniqueness of asymptotic cones in relatively hyperbolic groups, linking geometric properties to algebraic structures.

Contribution

It provides an explicit construction of the universal tree-graded space and proves the uniqueness of asymptotic cones under certain conditions, advancing understanding of geometric group theory.

Findings

Universal tree-graded spaces are explicitly constructed.

Asymptotic cones of certain groups are shown to be unique.

Homogeneous asymptotic cones with cut points are characterized as universal tree-graded spaces.

Abstract

We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique asymptotic cones. Modulo the Continuum Hypothesis, we show that if an asymptotic cone of a geodesic metric space is homogeneous and has cut points, then it is the universal tree-graded space with pieces - maximal connected subsets without their own cut points. Thus it is completely determined by its collection of pieces.