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Oberseminar Topology/Geometry
# Large-scale geometry of the Rips filtration

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SZ (MI)
### SZ

#### MI

Description

Given a metric space $X$ and a scale parameter $\sigma \geq 0$, the Rips graph $Rips^\sigma X$ has $X$ as its vertex set, with two vertices declared adjacent whenever their distance is at most $\sigma$. A classical fact is that $X$ is a quasigeodesic space precisely if it is quasi-isometric to its Rips graph at sufficiently large scale. By considering all possible scales, we obtain a directed system of graphs known as the Rips filtration. How does the large-scale geometry of $Rips^\sigma X$ evolve as $\sigma \to \infty$? Is there a meaningful notion of limit? It turns out that the answers depends on whether we work up to quasi-isometry or coarse equivalence. In this talk, I will discuss some results inspired by these questions.