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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)...

Suppose that $(X,d_X)$ and $(Y,d_Y)$ are complete doubling metric spaces and let $f:X\rightarrow Y$ be a non-constant Lipschitz map. Then can does there exist a lsc function $\rho:(0,\infty)\...