随机3-流形的均匀模型与短曲线
Uniform models and short curves for random 3-manifolds
AI总结：
本文提出了两种在满足特定拓扑条件的3-流形上构造双曲度量的方法，以渐近概率1适用于随机Heegaard分裂，并利用Kleinian群形变理论证明了几何不变量的增长率和随机3-流形的算术性等结果，且不依赖几何化猜想。
摘要：
我们提供了两种在满足特定拓扑条件的具有Heegaard分裂的3-流形上构造双曲度量（hyperbolic metrics）的方法，这两种方法均以渐近概率1适用于随机Heegaard分裂。这些构造对所得度量提供了大量控制，使我们能够证明关于几何不变量（如直径和内射半径）的粗略增长率，以及随机3-流形族中算术性和可公度性的各种结果。例如，我们证明了随机Heegaard分裂的直径随相关随机游走的长度呈粗略线性增长。这些构造仅使用Kleinian群的形变理论工具，即我们不依赖于Perelman对几何化猜想的证明。特别地，我们给出了Maher关于随机3-流形是双曲的这一结果的证明，该证明绕过了几何化。
英文摘要：
We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splittings with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results about the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and about arithmeticity and commensurability in families of random 3-manifolds. For example, we show that the diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk. The constructions only use tools from the deformation theory of Kleinian groups, that is, we do not rely on the solution of the Geometrization Conjecture by Perelman. In particular, we give a proof of Maher's result that random 3-manifolds are hyperbolic that bypasses Geometrization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
