A Combinatorial Criterion and Center for the quasi-isometry groups of Euclidean spaces

TL;DR

This paper introduces a new combinatorial criterion for identifying quasi-isometries of Euclidean spaces and proves that the center of the quasi-isometry group of  Euclidean space is trivial.

Contribution

It presents a novel combinatorial criterion based on simplicial structures to determine quasi-isometries and establishes the triviality of the center of the quasi-isometry group of  Euclidean space.

Findings

Introduced the concept of $PL_\delta$-homeomorphisms.

Provided a combinatorial criterion for quasi-isometries.

Proved the center of $QI(\u00a0bR^n)$ is trivial.

Abstract

In this study, we introduce the notion of $PL_\delta$-homeomorphisms of $\mathbb{R}^n$. Furthermore, we provide a combinatorial criterion reliant on the vertices and edges of simplicial structures, to determine whether a piecewise-linear homeomorphism to be a quasi-isometry. By employing this criterion, we subsequently show that the center of the group $QI(\mathbb{R}^n)$, which comprises all quasi-isometries of $\mathbb{R}^n$, is indeed trivial.