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

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 $$H=\{[f]\in QI({\mathbb{R}}^n):\underset{\mathrm{\parallel }x\mathrm{\parallel }\to \mathrm{\infty }}{\lim }\frac{\mathrm{\parallel }f(x)-x\mathrm{\parallel }}{\mathrm{\parallel }x\mathrm{\parallel }}=0\},$$ and for $0<\alpha <1$, $$H_{\alpha }=\{[f]\in QI({\mathbb{R}}^n):\mathrm{\parallel }f(x)-x\mathrm{\parallel }\le K\mathrm{\parallel }x{\mathrm{\parallel }}^{\alpha }\text{ for large }\mathrm{\parallel }x\mathrm{\parallel }\},$$ we show that each $H_{\alpha }$ is a nontrivial normal subgroup of $QI({\mathbb{R}}^n)$, satisfying $$H_{\alpha }\subset H_{\beta }\subset H\text{for }0<\alpha <\beta <1.$$ We prove that the centers of $QI({\mathbb{R}}^n)\mathrm{/}H$ and $QI({\mathbb{R}}^n)\mathrm{/}{H}_{\alpha }$ 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 $H$.