2017/07/05 by Stefan Wenger, Wenger, Stefan
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #math.DG #math.GR #math.MG
paper · pdf · doi:10.48550/arxiv.1707.01398
Added Theorems 1.3, 7.1, and 7.2 which provide "bounded-scale" and "coarse" analogs of the previous main theorem. Slightly changed title to reflect the fact that the results also apply to bounded scales
arxiv created 2018/11/08 · arxiv updated 2018/11/09
We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are \rm CAT(0). This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper \rm CAT(0) in terms of the growth of the Dehn function at all scales. We finally obtain a generalization of this result of Lytchak and the author. Namely, we show that if the Dehn function of a proper, geodesic metric space is sufficiently close to the Euclidean Dehn function up to some scale then the space is not far (in a suitable sense) from being \rm CAT(0) up to that scale.