Large-scale geometry of the Rips filtration

Given a metric space X and a scale parameter σ ≥ 0, the Rips graph RipsσX has X as its vertex set, with two vertices declared adjacent whenever their distance is at most σ. 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 RipsσX evolve as σ → ∞? 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 and applications inspired by these questions.