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Let us phrase the question in the title in more detail: I wonder if there exists a metric space $X$ which has at least two points, has finite diameter (in the sense that there is an upper bound for th …

A consequence of the famous Jørgensen inequality is that there is a lower bound for the distance between closed geodesics in hyperbolic three-manifolds: for any $R>0$ there is a c>0 such that for any …

Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other "repelling". …

I'm not familiar with the Götsky--Cohn construction but Siegel's (as explained in van der Geer's book) seems clear: there is a "height function" $y$ on $X = \mathbb H^2 \times \mathbb H^2$ (the "di …

An answer was given in Ashot Minasyan's comment, i'm writing it here so the question does not remain unanswered. In Lemma 2.5 of "Group actions on metric spaces: fixed points and free subgroups" by Ma …

This is only an answer to one point of your question: for surfaces of large genus $g$ the distance should be $$ d(S, \mathrm{point}) \asymp \log(g). $$ The lower bound should follow from volume esti …

In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality $$ \ell \ge 2\pi \epsilon^{-1} $$ which follows from the fact …