Orderability and Asymptotic Structure of $\mathrm{QI}(\mathbb{R}^n)$

In this article, we study the algebraic and dynamical structure of certain normal subgroups of the quasi-isometry group of Euclidean spaces.

We first consider the normal subgroup consisting of quasi-isometries that are asymptotically equal to the identity, and introduce a nested family of normal subgroups that distinguish different orders of sublinear deviation from the identity.

We show that the centers of the resulting quotient groups are trivial.

We further prove that these quotient groups are neither left-orderable nor locally indicable.

We also introduce an asymptotic topology on the quasi-isometry group, yielding a natural metric structure on the quotient and providing a framework for studying large-scale invariants.