Collapsing Maps and Quasi-Isometries

TL;DR

This paper introduces a new class of metrics called (b,c)-metrics, explores their properties under quasi-isometries, and defines collapsing maps that preserve quasi-isometric structure in these spaces.

Contribution

It generalizes the concept of b-metrics to (b,c)-metrics and demonstrates that certain collapsing maps are quasi-isometries within this framework.

Findings

(b,c)-metric spaces are preserved under quasi-isometries

Collapsing maps can be constructed as quasi-isometries

The (b,c)-metric framework generalizes b-metrics

Abstract

We introduce a generalization of the b-metric we call a (b,c)-metric. We show that if $X$ is a $(b,c)$-metric space and $\psi: X \longrightarrow Y$ is a quasi-isometry then $Y$ is $(b,c)$-metrizable. We also define a particular kind of collapsing map that can be applied to an arbitrary $(b,c)$-metric space. We define a distance function on the image of this collapsing map and with this prove that the collapsing map is a quasi-isometry.