Given a metric space $(X, d),$ the set of all self-isometries $f \colon (X, d) \to (X, d)$ forms a group $$\text{Isom}(X)$$ under composition.
Let $$\text{Isom}_{0}(X) = \left\{f \in \text{Isom}(X) \mid \sup_{x \in X} d(f(x), x) < \infty\right\}$$ denote the subset of $\text{Isom}(X)$ consisting of self-isometries of $X$ which are a bounded distance away from the identity $\text{id}_{X}.$
Working through the definitions, one can show that $\text{Isom}_{0}(X)$ is a normal subgroup of $\text{Isom}(X),$ hence we can form the quotient group $$ \text{Isom}(X) /\text{Isom}_{0}(X).$$ Note that two isometries $f, g \in \text{Isom}(X)$ are mapped to the same element of $\text{Isom}(X) /\text{Isom}_{0}(X)$ by the natural projection if and only if $$\sup_{x \in X}d(f(x), g(x)) < \infty.$$
I was motivated to consider this construction because of the similar quotienting process used to construct the quasi-isometry group $\mathcal{QI}(X)$ of a metric space. My question is: Are there any papers/books which discuss this quotient group?