Centers and Orderability of Certain Quotients of quasi-isometry groups of Euclidean spaces

TL;DR

This paper investigates the algebraic and dynamical properties of specific normal subgroups within the quasi-isometry group of Euclidean spaces, revealing their structure, centers, and topological density.

Contribution

It introduces and analyzes the subgroups $H_eta$ of $QI(R^n)$, establishing their normality, inclusion relations, and topological density, and examines the properties of their quotient groups.

Findings

Each $H_eta$ is a nontrivial normal subgroup of $QI(R^n)$.

The centers of the quotient groups are trivial, but they contain nontrivial torsion elements.

The subgroups $H_eta$ are dense in $H$ under the asymptotic topology.

Abstract

In this article, we study the algebraic and dynamical structure of certain normal subgroups of the quasi-isometry group of Euclidean space $QI(\mathbb{R}^n)$. For \[ H = \Big\{ [f] \in QI(\mathbb{R}^n) : \lim_{\|x\|\to\infty} \frac{\|f(x)-x\|}{\|x\|} = 0 \Big\}, \] and for $0<\alpha<1$, \[ H_\alpha = \Big\{ [f] \in QI(\mathbb{R}^n) : \|f(x)-x\| \le K\|x\|^\alpha \text{ for large } \|x\| \Big\}, \] we show that each $H_\alpha$ is a nontrivial normal subgroup of $QI(\mathbb{R}^n)$, satisfying \[ H_\alpha \subset H_\beta \subset H \qquad \text{for } 0<\alpha<\beta<1. \] We prove that the centers of $QI(\mathbb{R}^n)/H$ and $QI(\mathbb{R}^n)/H_\alpha$ are trivial, while these quotients admit nontrivial torsion elements. Consequently, they are neither left-orderable nor locally indicable. Finally, we introduce an asymptotic topology on $QI(\mathbb{R}^n)$ and show that the family $\{H_\alpha\}_{0<\alpha<1}$ is dense in $H$.