Markoff Surfaces and Superstrong Approximation

无理数的存在自古以来就吸引着人类；用有理数逼近它们在理论和应用上都具有重要意义。马可夫丢番图方程出现在马可夫的基本工作中（1879年），描述了那些非常近似的无理数。这个方程出现在几个数学领域。它的整数解具有树形结构，对这些马尔科夫数的算术性质的研究（其中第一个与黄金比例有关，在某种意义上是最糟糕的近似无理数）导致了图论中的重要问题。本研究计划探讨与Markoff树相关的图的连通性。除了对数论的重要性外，本研究还与产品替换算法（计算群论中最普遍但仍知之甚少的工具）和计算机科学中的扩展图等主题有着深刻的联系和应用。对Markoff数的算术性质的研究导致了由Markoff树的模约化得到的图是否连通（强逼近）的问题。超强近似是断言这些图实际上是高度连接的，也就是说形成了一个展开族。在过去的十年中，在瘦群（无限指数的Zariski密集子群）的超强近似领域出现了惊人的爆发。许多研究都是由仿射筛的发展推动的。对于具有Levi因子的Zariski闭包半单质的薄群，强近似和超强近似及其在仿射筛中的应用现在已经得到了很好的理解。另一方面，环面提出了特别困难的问题，无论是在轨道上元素的稀疏性和它们的丢芬图性质方面，还是在强近似方面，在这种情况下，强近似相当于Artin的原始根猜想。本研究项目将研究在环面和细线性群之间的中等难度设置中的强逼近和超强逼近，即由Markoff方程定义的曲面上的非线性作用，以及其他Markoff类型曲面的背景。细线性群的超强逼近结果将在本研究中发挥重要作用，与Lang猜想的进展和代数painlevevlevi方程的分类有关的技术和方法也将发挥重要作用。

The existence of irrational numbers has fascinated mankind since antiquity; their approximation by rational numbers is of great importance in both theory and applications. The Markoff Diophantine equation arose in Markoff's fundamental work (in 1879) characterizing those irrationals that are badly approximable. The equation arises in several fields of mathematics. Its integer solutions have a tree structure, and investigation of the arithmetic properties of these Markoff numbers (the first of which is associated with the golden ratio, in a sense the most badly approximable irrational number) leads to important questions in graph theory. This research project investigates the connectivity of graphs related to the Markoff tree. In addition to its importance for number theory, this investigation has deep connections and applications to, among other topics, the product replacement algorithm (the most prevalent but still poorly understood tool in computational group theory) and expander graphs in computer science.Investigation of the arithmetic properties of Markoff numbers leads to the question of whether the graphs obtained by the modular reduction of the Markoff tree are connected (strong approximation). Superstrong approximation is the assertion that these graphs are in fact highly connected, that is to say form a family of expanders. The past decade saw a remarkable explosion of activity in the area of superstrong approximation for thin groups (Zariski dense subgroups of infinite index). Much of the research was driven by development of the affine sieve. In the case of thin groups with Levi factor of its Zariski closure semisimple, the strong and superstrong approximation and their applications in affine sieve are by now well-understood. On the other hand, the tori pose particularly difficult problems, both in terms of sparsity of elements in an orbit and their Diophantine properties as well as in terms of strong approximation, which in this case amounts to Artin's primitive root conjecture. This research project will investigate strong and superstrong approximation in a setting that is intermediate in level of difficulty between that of tori and that of thin linear groups, namely, that of nonlinear actions on a surface defined by the Markoff equation as well as in the context of other surfaces of Markoff type. Superstrong approximation results for thin linear groups will play an important role in this investigation, as will techniques and methods related to progress on Lang's conjecture and to the classification of algebraic Painlevé VI equations.