Large-scale geometry of the Rips filtration

by Robert Tang

Tuesday 18 Jun 2024, 16:00 → 18:00 Europe/Zurich

SZ (MI)

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.