We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex of the graph. This extends previously established results for the pants graph and the separating curve graph to a broad family of graphs associated to surfaces.

中文翻译：

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多曲线图的厚度和相对双曲线

我们证明任何满足某些自然属性的多曲线图要么是双曲线的，要么是相对双曲线的，要么是厚的。此外，这种几何特征是由与图的每个顶点相交的一组子表面确定的。这将先前建立的裤子图和分离曲线图的结果扩展到与表面相关的广泛的图族。