论文标题
双曲线三角形和曾经是圆环组的形状
Shapes of hyperbolic triangles and once-punctured torus groups
论文作者
论文摘要
令$δ$为具有固定面积$φ$的双曲线三角形。我们证明，对于所有$δ$的通用选择，对于所有$δ$的通用选择而言，$π$产生的属性 - 围绕三角形侧面的中点旋转，没有承认非平凡的关系。相比之下，我们显示了所有$φ\ in（0，π）\ setMinus \ mathbb {q}π$，一组密集的三角形确实提供了非平凡关系，在通用案例映射到双曲线翻译中。为了确定这一事实，我们研究了一个圆锥形$θ= 2（π-φ）$的奇异双曲线指标的变形空间$ \ mathfrak {c}_θ$，并回答一个多重质量结构$ξ$ $ρ_$ρ_P的类似问题。在X.的附录中，给出了$θ$的具体示例和$ Mathfrak in \ Mathfrak {c}_θ$的具体示例，而每个$ρ_PON$的图像有限地呈现，不免费和免费扭转；实际上，这些图像对于封闭的双曲线3 - manifolds的基本组将是同构。
Let $Δ$ be a hyperbolic triangle with a fixed area $φ$. We prove that for all but countably many $φ$, generic choices of $Δ$ have the property that the group generated by the $π$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $φ\in(0,π)\setminus\mathbb{Q}π$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_θ$ of singular hyperbolic metrics on a torus with a single cone point of angle $θ=2(π-φ)$, and answer an analogous question for the holonomy map $ρ_ξ$ of such a hyperbolic structure $ξ$. In an appendix by X.~Gao, concrete examples of $θ$ and $ξ\in\mathfrak{C}_θ$ are given where the image of each $ρ_ξ$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds.
