## Metric spaces with unique and uniformly close tangents and their metric dimensions

#### Enrico Le DonneUniversity of Jyväskylä

In this talk we will consider metric spaces that at every point admit a tangent metric space and moreover the convergence toward the tangent cones is uniform. Familiar examples are Reifenberg vanishing flat metric spaces and equiregular subRiemannian manifolds. I will explain why such tangents are homogeneous spaces (almost everywhere, when a doubling measure is present). In case of doubling geodesic spaces, these tangents are actually well-known objects: Carnot groups. I will present some new results, done in collaboration with U. Lang and T. Rajala, regarding metric dimensions. Namely, if all tangent cones of a metric space with unique and uniformly close tangents have (Assouad or Nagata) dimension n, with uniform constant, then the space has dimension n, locally. In particular, equiregular subRiemannian manifolds have locally controlled dimension equal to the topological one.

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