Hyperbolic Dimension and Decomposition Complexity

TL;DR

This paper introduces new tools and equivalent definitions for decomposition complexity, linking it to hyperbolic dimension and coarse embeddability, and discusses methods to identify spaces lacking finite decomposition complexity.

Contribution

It establishes three equivalent definitions for decomposition complexity and connects it to hyperbolic dimension and coarse embeddability into Hilbert space.

Findings

Spaces with finite hyperbolic dimension have finite decomposition complexity

The class of spaces coarsely embeddable into Hilbert space is closed under decomposition

Methods are provided to show certain spaces lack finite decomposition complexity

Abstract

The aim of this paper is to provide some new tools to aid the study of decomposition complexity, a notion introduced by Guentner, Tessera and Yu. In this paper, three equivalent definitions for decomposition complexity are established. We prove that metric spaces with finite hyperbolic dimension have finite (weak) decomposition complexity, and we prove that the collection of metric families that are coarsely embeddable into Hilbert space is closed under decomposition. A method for showing that certain metric spaces do not have finite decomposition complexity is also discussed.