Let $X$ be a connected compact metric space.


**Question:** Is there a sequence of compact hyperbolic surfaces (the curvature may differ between surfaces) that converges to $X$ in the Gromov-Hausdorff metric?



Can hyperbolic surfaces approximate every connected compact metric space?
The answer is "yes".
Moreover, given an integer $n\geqslant 2$, there a sequence of compact hyperbolic $n$-manifolds that converges to $X$ in the sense of Gromov--Hausdorff.


See V. Šahović. “Approximations of Riemannian manifolds with linear curvature constraints.” Thesis. Univ. Münster, 2009.


