论文标题
计算Veech组的规范翻译表面
Canonical translation surfaces for computing Veech groups
论文作者
论文摘要
对于翻译表面空间的每个阶层，我们以适当的方式引入了一个无限的翻译表面，该表面的每个翻译表面的副本副本。给定层中的翻译表面$（x，ω）$，当时矩阵位于其VEECH组$ \ mathrm {sl}（x，ω）$时，并且仅当时仅当相关的无限表面的相关仿射自动形态发送有限的集合，每一个有限的集合，标记为``````到Voronoi 1细胞，到另一对方向对``标记''段。 我们证明了独立利益的结果。对于每个真正的$ a \ ge \ sqrt {2} $，都有一个明显的双曲球，以至于对于任何富赫西亚的群体琐碎稳定$ i $，该组的$ i $中心已经以$ i $在球内同意的dirichlet域，该小组与frobenius norm norm norm norm uns norm um a frobolic elements uns norm a $ a $ a a $ a n of the Mallobolic Half-Planes的相交％当$ \ mathrm {sl}（x，ω）$是一个晶格时，我们会使用它来保证已计算完整的组$ \ mathrm {sl}（x，ω）$。 总之，这些结果引起了用于计算Veech组的新算法。
For each stratum of the space of translation surfaces, we introduce an infinite translation surface containing in an appropriate manner a copy of every translation surface of the stratum. Given a translation surface $(X, ω)$ in the stratum, a matrix is in its Veech group $\mathrm{SL}(X,ω)$ if and only if an associated affine automorphism of the infinite surface sends each of a finite set, the ``marked" {\em Voronoi staples}, arising from orientation-paired segments appropriately perpendicular to Voronoi 1-cells, to another pair of orientation-paired ``marked" segments. We prove a result of independent interest. For each real $a\ge \sqrt{2}$ there is an explicit hyperbolic ball such that for any Fuchsian group trivially stabilizing $i$, the Dirichlet domain centered at $i$ of the group already agrees within the ball with the intersection of the hyperbolic half-planes determined by the group elements whose Frobenius norm is at most $a$. %When $\mathrm{SL}(X,ω)$ is a lattice we use this to give a condition guaranteeing that the full group $\mathrm{SL}(X,ω)$ has been computed. Together, these results give rise to a new algorithm for computing Veech groups.
