Topology Seminar

Wednesday, September 28, 2016
4:20 PM - 5:30 PM (ET)
Exley Science Center Tower ESC 638
Event Type
Canalia, Caryn

Dave Constantine, Wes: "Hausdorff dimension and the CAT(K) condition for surfaces"

Abstract: A geodesic metric space satisfies the CAT(K) condition if its geodesic triangles are all `thinner' than triangles with the same side lengths in the model space of constant Riemannian curvature K. This condition allows one to extend many arguments relying on an upper curvature bound from Riemannian geometry to the metric space setting.


How `strange' can a metric be while still satisfying the CAT(K) property? One way to measure this is with the difference between the topological dimension of the space and its Hausdorff dimension with respect to the metric. In this talk I'll show that, at least for surfaces, a CAT(K) metric is tame in the sense that it yields Hausdorff dimension 2. I'll also provide some motivation for this question by showing how results like this allow one to extend volume entropy rigidity statements to the CAT(-1) setting. 

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