A combinatorial criterion and center for the quasi-isometry groups of Euclidean spaces

Abstract

In this study, we introduce the notion of PLδ-homeomorphisms of Rn. 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(Rn), which comprises all quasi-isometries of Rn, is indeed trivial.