Recently, I came across the notion of [quasi-isometries](https://en.wikipedia.org/wiki/Quasi-isometry), 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?




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**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](https://mathoverflow.net/questions/288909/does-there-exist-a-notion-of-discrete-riemannian-metric-on-graph)). In that post, the described graphs are quasi-isometrically embedded into the target manifolds.



What properties are preserved by quasi-isometries
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*.


