Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine interval exchange maps. However, though the $$SL(2,\mathbb {R})$$ S L ( 2 , R ) -action is ergodic in connected components of strata of translation surfaces, there may be mutually disjoint $$SL(2,\mathbb {R})$$ S L ( 2 , R ) -invariant open subsets in components of dilation surfaces. A first distinction is between triangulable and non-triangulable dilation surfaces. For non-triangulable surfaces, the action of $$SL(2,\mathbb {R})$$ S L ( 2 , R ) is somewhat trivial so the study can be focused on the space of triangulable dilation surfaces. In this paper, we introduce the notion of horizon saddle connections in order to refine the distinction between triangulable and non-triangulable dilation surfaces. We also introduce the family of quasi-Hopf surfaces that can be triangulable but display the same trivial behavior as non-triangulable surfaces. We prove that existence of horizon saddle connections drastically restricts the Veech group a dilation surface can have.

中文翻译：

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Horizo​​n 鞍点连接、准 Hopf 曲面和 Veech 膨胀曲面群

膨胀面是平移面的推广，其中几何结构在复平面上建模到线性部分为实数的仿射图。它们是研究仿射区间交换图悬浮的几何框架。然而，尽管 $$SL(2,\mathbb {R})$$ SL ( 2 , R ) -action 在平移面层的连通分量中是遍历的，但可能存在相互不相交的 $$SL(2,\mathbb {R})$$ SL ( 2 , R ) - 膨胀表面分量中的不变开放子集。第一个区别是可三角化和不可三角化的膨胀表面。对于不可三角曲面，$$SL(2,\mathbb {R})$$SL ( 2 , R ) 的作用有些微不足道，因此可以将研究重点放在可三角膨胀曲面的空间上。在本文中，我们引入了地平线鞍形连接的概念，以细化可三角化和不可三角化膨胀表面之间的区别。我们还介绍了准 Hopf 曲面系列，它们可以是可三角化的，但表现出与不可三角化的曲面相同的微不足道的行为。我们证明地平线鞍形连接的存在极大地限制了膨胀表面可以具有的 Veech 群。