Geometry and computability in low-dimensional topology and group theory.

在瑟斯顿几何猜想的证明下，3-流形在某种意义上被分类。然而，证明的非构造性给出了判定两个给定的3-流形是否相同的直接方法。在这个项目中，Ascari将研究这些计算问题。他将使用几种技术来接近它。一方面，诸如层次结构之类的拓扑工具将被证明是有用的。另一方面，代数方法，例如使用基础群的无限完备化，也将是卓有成效的。在这两种方法之上的是双曲几何的普遍作用，这是Ascari的专业领域。这些问题特别令人感兴趣的一个领域是纽结理论，阿斯卡里也将研究这一领域。我们现在更详细地解释这些方法。有限生成群的有限完备化是一个紧拓扑群，它是它的有限商的逆极限。我们猜想，三维双曲流形的基本群的无限完备性应该完全决定这个流形。如果这是真的，这将为3-流形的同胚问题提供一个新的解决方案。达里奥的两位主管都有这方面的专业知识。Lackenby用有限完备性研究了三维流形的有限薄片覆盖，Bridson最近发现了由其有限完备性决定的双曲三维流形的第一个例子。实际上，它是由它在所有有限生成的剩余有限群中的超定完备性决定的。3-流形的层次是沿着不可压缩曲面的有限分解序列，它将流形切割成3-球。Haken证明了许多3-流形都有一个层次，特别是所有纽结补都有一个层次。他用这些方法给出了节点和链节的等价问题的第一个解决方案。Lackenby使用层次结构对这一问题和相关问题的计算复杂性产生了定量的界限。例如，他证明了识别结点的问题在于复杂类co-np。在他的项目中，达里奥将使用这些方法来分析更复杂的结。这将与对无限完备化的研究相一致，因为证明无限刚性似乎需要以某种方式使用不可压缩曲面。这个项目将利用达里奥的主管Lackenby和Bridson的许多不同领域的专业知识。它将需要在低维拓扑学、双曲几何和几何群论方面的复杂方法。该项目位于EPSCR研究领域几何与拓扑学和代数。

With the proof of Thurston's Geometrisation Conjecture, 3-manifolds are in some sense classified. However, the non-constructive nature of the proof gives no direct method for deciding whether two given 3-manifolds are the same. In this project, Ascari will examine these computational questions. He will approach it using several techniques. On the one hand, topological tools such as hierarchies will prove to be useful. On the other hand, algebraic approaches, such as the use of the profinite completion of the fundamental group, will also be fruitful. Overlying both these approaches is the pervasive role of hyperbolic geometry, which is an area of expertise of Ascari. A field where these questions are particularly of interest is knot theory, which Ascari will also examine. We now explain these approaches in a bit more detail. The profinite completion of a finite generated group is a compact topological group that is the inverse limit of its finite quotients. It is conjectured that the profinite completion of the fundamental group of a hyperbolic 3-manifold should completely determine the manifold. If true, this would provide a new solution to the homeomorphism problem for 3-manifolds. Both of Dario's supervisors have expertise in this area. Lackenby has used profinite completions to study finite-sheeted covers of 3-manifolds, and Bridson has recently discovered the first example of a hyperbolic 3-manifold that is determined by its profinite completion. In fact, it is determined by its profinite completion among all finitely generated residually finite groups. A hierarchy for a 3-manifold is a finite sequence of decompositions along incompressible surfaces that cuts the manifold into 3-balls. Haken showed that many 3-manifolds have a hierarchy; in particular, all knot complements have one. He used these to produce the first solution to the equivalence problem for knots and links. Lackenby has used hierarchies to produce quantitative bounds on the computational complexity of this and related problems. For example, he showed that the problem of recognising the unknot lies in the complexity class co-NP. In his project, Dario will use these methods to analyse more complicated knots. This will tie in with the study of profinite completions, since it seems likely that a proof of profinite rigidity will need to use incompressible surfaces in some way. This project will draw on many different areas of expertise of Dario's supervisors Lackenby and Bridson. It will require sophisticated methods in low-dimensional topology, hyperbolic geometry and geometric group theory. The project lies in the EPSCR Research Areas Geometry & Topology and Algebra.