In his seminal work on finitely generated groups, Gromov established that such groups possess a well-defined large-scale metric structure induced by the word metric of a finite generating set. This perspective transformed geometric group theory by introducing quasi-isometric invariants, a prominent example of which is the asymptotic dimension—a large-scale analogue of the Lebesgue covering dimension. A parallel milestone in this geometric framework was the proof of Gromov’s polynomial growth theorem, which characterises groups with polynomial growth, utilising the Gromov–Hausdorff distance to quantify dissimilarities between metric spaces.

Almost a decade later, Topological Data Analysis (TDA), a field at the interplay of computational geometry, computer science, and algebraic topology, emerged to study the shape of data. The main tools are topology-inspired invariants, such as persistent homology, used to extract geometric features from datasets. Within this framework, both classic metric notions found new, independent utilities. The Gromov–Hausdorff distance became a standard tool for comparing datasets and evaluating the stability of invariants. The asymptotic dimension was used to analyse the spaces of these invariants, thereby bounding the unavoidable information loss incurred during their vectorisation, a necessary step to integrate them into statistical and machine learning pipelines.

In this talk, we discuss how the asymptotic dimension and the Gromov–Hausdorff distance, originally introduced in the realm of topological methods to study algebraic structures, have gained a crucial role in TDA, and present recent results that bridge these notions by determining the asymptotic dimension of the Gromov–Hausdorff space.

N. Zava, Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces, Algebraic & Geometric Topology 25 (8), 5153-5174.