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Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones".

What (metric)/geometric properties are preserved by quasi-isometries? Also, are there good references on the topic?

Direction/Angle: Concretely, as an example of the direction I'm thinking in, I am interested in graph approximations to compact smooth manifolds (e.g. in this post). In that post, the described graphs are quasi-isometrically embedded into the target manifolds.

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    $\begingroup$ This is a very broad question... and at the same time I don't really see how the linked post is related to the question. $\endgroup$
    – YCor
    Commented Nov 29, 2022 at 23:26
  • $\begingroup$ Ah the graph is quasi symmetrically embedded into that manifold. $\endgroup$
    – ABIM
    Commented Nov 30, 2022 at 1:52
  • $\begingroup$ If you're interested in "graph approximations to compact smooth manifolds", then I'm not certain that quasi-isometries (or, at least, much of the literature on them) is going to be relevant to you. Every compact smooth manifold is quasi-isometric to a point. What is true is that, if you want to approximate a compact manifold by a fine graph "mesh", then the quality of a $(\lambda,\epsilon)$-quasi-isometry will improve as the mesh gets finer -- that is, $(\lambda,\epsilon)\to (1,0)$. So the quality of the quasi-isometry should be important in your setting. $\endgroup$
    – HJRW
    Commented Nov 30, 2022 at 13:56
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    $\begingroup$ @HJRW Yes indeed, I am most curious about some quasi-isometries with minimal distortion ($\lambda$) small $\epsilon$ quasi-isometric. Thanks for pointing this out :) $\endgroup$
    – ABIM
    Commented Nov 30, 2022 at 14:32

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One can think of quasi-isometric spaces as spaces which look the same when seen from far away. Examples of properties preserved under quasi-isometries are for example Gromov-hyperbolicity (for geodesic metric spaces), growth types of Dehn functions and various notions of "rank". As a reference I would recommend Buyalo-Schroeder: Elements of Asymptotic Geometry, Bridson-Haefliger: Metric spaces of non-positive curvature and Burago-Burago-Ivanov: A course in metric spaces.

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