曲线图的模型论
The model theory of the curve graph
AI总结：
本文从模型论视角研究有限型曲面的曲线图，通过其与映射类群增广凯莱图的双向解释性，证明其理论为ω-稳定、有量词消去等性质，提出解释刚性现象，搭建了模型论与几何拓扑、群论的桥梁。
摘要：
本文在模型论、几何拓扑与几何群论之间搭建了桥梁。具体而言，我们从模型论的视角研究了伊万诺夫元猜想（Ivanov Metaconjecture），并更广泛地试图回答一个一般性问题：为何曲面的曲线图（curve graph）在曲面与映射类群的研究中占据如此核心的地位？\n 更具体地，我们考虑有限型曲面$\Sigma$及其曲线图$\mathcal C(\Sigma)$，并在图论语言下研究其一阶理论。关键在于，$\mathcal C(\Sigma)$与一个被称为该曲面映射类群的增广凯莱图（augmented Cayley graph）的特定对象是双向可解释的。我们利用这一双向解释性证明了曲线图的理论是$\omega$--稳定的，计算了其莫利秩（Morley rank），并证明它相对于$\forall\exists$--公式类具有量词消去性质。我们还证明了许多自然关联于曲面的复形都可在$\mathcal C(\Sigma)$中得到解释，这表明这些复形均为$\omega$--稳定的，且其莫利秩存在特定的先验界。我们能够利用莫利秩证明各类复形与曲线图不具备双向可解释性。作为量词消去的一个推论，我们证明了代数相交数在曲线图的一阶理论中是不可定义的。最后，我们证明曲面的曲线图具有一种我们称之为“解释刚性”的新现象：即若曲面$\Sigma_1$与$\Sigma_2$的曲线图是相互可解释的，则$\Sigma_1$与$\Sigma_2$彼此同胚。研究过程中还得到了大量技术性结果。
英文摘要：
In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. In particular, we investigate the Ivanov Metaconjecture from the point of view of model theory, and more broadly we seek to answer the general question: why does the curve graph of a surface play such a central role in the study of surfaces and mapping class groups? More specifically, we consider a surface $Σ$ of finite type and its curve graph $\mathcal C(Σ)$, and we investigate its first-order theory in the language of graph theory. Crucially, $\mathcal C(Σ)$ is bi-interpretable with a certain object called the augmented Cayley graph of the mapping class group of the surface. We use this bi-interpretation to prove that the theory of the curve graph is $ω$--stable, to compute its Morley rank, and to show that it has quantifier elimination with respect to the class of $\forall\exists$--formulae. We also show that many of the complexes which are naturally associated to a surface are interpretable in $\mathcal C(Σ)$. This shows that these complexes are all $ω$--stable and admit certain a priori bounds on their Morley ranks. We are able to use Morley ranks to prove that various complexes are not bi--interpretable with the curve graph. As a consequence of quantifier elimination, we show that algebraic intersection number is not definable in the first order theory of the curve graph. Finally, we prove that the curve graph of a surface enjoys a novel phenomenon that we call interpretation rigidity. That is, if surfaces $Σ_1$ and $Σ_2$ admits curve graphs that are mutually interpretable, then $Σ_1$ and $Σ_2$ are homeomorphic to each other. Along the way, numerous technical results are obtained.
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