Given a metric space $X$, is there a natural way to view the quasi-isometry group $QI(X)$ as a topological group?

Question

Given a metric space $(X,d)$, we define $QI(X)$ as the set of quasi-isometries $f : X \to X$, modulo the equivalence relation $$ f \sim g \ \ \ \ \text{ if and only if } \ \ \ \sup_{x \in X} \ d(f(x),g(x)) < \infty. $$ This set admits an obvious group structure via composition.

The group of isometries $Isom(X)$ is often viewed as a topological group. My question is then: is there ever a natural way to equip $QI(X)$ with a topology, perhaps for specific 'nice' $X$?

Thanks