A quantitative version of the Morse lemma and quasi-isometries fixing the ideal boundary

The Morse lemma is fundamental in hyperbolic group theory. Using exponential contraction, we establish an upper bound for the Morse lemma that is optimal up to multiplicative constants, which we demonstrate by presenting a concrete example. We also prove an “anti” version of the Morse lemma. We introduce the notion of a geodesically rich space and consider applications of these results to the displacement of points under quasi-isometries that fix the ideal boundary.