22,358 research outputs found

    Triangulations and volume form on moduli spaces of flat surfaces

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    In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some deformation of the moduli space of translation surfaces. Using geodesic triangulations, we define a volume form on this moduli space, and show that, in the well-known cases, this volume form agrees with usual ones, up to a multiplicative constant.Comment: 42 page

    Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps

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    We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.Comment: 27pgs No figs, v3: final version, incorporating referee's comments, to appear in Geometriae Dedicat

    Gromov-Monge quasi-metrics and distance distributions

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    Applications in data science, shape analysis and object classification frequently require maps between metric spaces which preserve geometry as faithfully as possible. In this paper, we combine the Monge formulation of optimal transport with the Gromov-Hausdorff distance construction to define a measure of the minimum amount of geometric distortion required to map one metric measure space onto another. We show that the resulting quantity, called Gromov-Monge distance, defines an extended quasi-metric on the space of isomorphism classes of metric measure spaces and that it can be promoted to a true metric on certain subclasses of mm-spaces. We also give precise comparisons between Gromov-Monge distance and several other metrics which have appeared previously, such as the Gromov-Wasserstein metric and the continuous Procrustes metric of Lipman, Al-Aifari and Daubechies. Finally, we derive polynomial-time computable lower bounds for Gromov-Monge distance. These lower bounds are expressed in terms of distance distributions, which are classical invariants of metric measure spaces summarizing the volume growth of metric balls. In the second half of the paper, which may be of independent interest, we study the discriminative power of these lower bounds for simple subclasses of metric measure spaces. We first consider the case of planar curves, where we give a counterexample to the Curve Histogram Conjecture of Brinkman and Olver. Our results on plane curves are then generalized to higher dimensional manifolds, where we prove some sphere characterization theorems for the distance distribution invariant. Finally, we consider several inverse problems on recovering a metric graph from a collection of localized versions of distance distributions. Results are derived by establishing connections with concepts from the fields of computational geometry and topological data analysis.Comment: Version 2: Added many new results and improved expositio
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