AI 中文总结
本文证明对具有自相似端空间且每个端由亏格累积的曲面，任意可数SL(2,R)子群可实现为膨胀曲面的Veech群，并可在指定方向实现水平鞍连接，与闭曲面情形形成对比。
AI 中文摘要
本文研究了基本群非有限生成的膨胀曲面的Veech群。我们证明，若$S$是具有自相似端空间且每个端都由亏格累积的曲面，则$SL(2,\mathbb{R})$的每个可数子群都可以实现为与$S$同胚的膨胀曲面的Veech群。此外，我们可以构造该膨胀曲面，使其在给定方向集合上实现水平鞍连接。这与闭膨胀曲面的情形形成对比，在闭膨胀曲面中，水平鞍连接对Veech群的代数结构施加限制。
英文摘要
This paper studies Veech groups of dilation surfaces whose fundamental group is not finitely generated. We prove that if $ S$ is a surface with self-similar end space and every end is accumulated by genus, then every countable subgroup of $SL(2,\mathbb{R}) $ can be realized as the Veech group of a dilation surface homeomorphic to $ S$. Additionally, we can construct this dilation surface such that it realizes horizon saddle connections in a prescribed set of directions. This contrasts with the case of closed dilation surfaces, where horizon saddle connections impose restrictions on the algebraic structure of the Veech group.
