Assouad-Nagata dimension of tree-graded spaces

TL;DR

This paper investigates the Assouad-Nagata dimension of tree-graded spaces, establishing a relation between the dimensions of the pieces and the entire space, and deriving a formula for free products of groups.

Contribution

It proves that the asymptotic Assouad-Nagata dimension of a tree-graded space can be bounded using the dimensions of its pieces, and provides a formula for free products of groups.

Findings

Dimension function for pieces extends to the whole space with a constant factor.

Asymptotic Assouad-Nagata dimension of free products equals the maximum of the factors.

The paper introduces a linear dimension function for tree-graded spaces.

Abstract

Given a metric space $X$ of finite asymptotic dimension, we consider a quasi-isometric invariant of the space called dimension function. The space is said to have asymptotic Assouad-Nagata dimension less or equal $n$ if there is a linear dimension function in this dimension. We prove that if $X$ is a tree-graded space (as introduced by C. Drutu and M. Sapir) and for some positive integer $n$ a function $f$ serves as an $n$-dimensional dimension function for all pieces of $X$, then the function $300\cdot f$ serves as an $n$-dimensional dimension function for $X$. As a corollary we find a formula for the asymptotic Assouad-Nagata dimension of the free product of finitely generated infinite groups: $asdim_{AN} (G*H)= max\{asdim_{AN} (G), asdim_{AN} (H)\}.$