Homomorphisms to hyperbolic and mapping class groups

一对(离散)群之间的同态集合是一个离散集合，近年来已被证明具有大量的结构。在这个项目中，PI将使用几何群论(作用于度量空间的群)的技术在两个场景中研究这个集合。第一种是当目标群体是双曲线群体时。在这种情况下，重点是算法，我们寻找显式描述同态集合的算法。这将导致对双曲群的逻辑进行算法研究。第二种情形是当目标群是有限类型可定向曲面S的映射类群时，如果B是一个好空间，则B上的S丛的(同构类)集自然由从B的基本群到S的映射类群的同态(同构类)来参数化。因此，对这组同态的研究对于曲面丛的研究是基本感兴趣的，曲面丛是数学中广泛关注的问题。群就是数学对象的对称集。把对称看作一种物理运动，一个人可以“什么都不做”，“完成一个对称，接着另一个对称”，以及“撤销对称”。这为对称集赋予了代数结构，而抽象这个概念则给出了一个群。这个项目将利用G是对称集合的空间的几何性质来研究群G。它将解决关于求解群上方程的算法的问题，这些问题的动机来自理论计算机科学和逻辑学。第二个重点是利用群论研究曲面丛。曲面丛是一个局部看起来像两个较低维空间(其中一个是二维空间)的乘积的空间，它们在整个纯数学中出现。

The set of homomorphisms between a pair of (discrete) groups is a discrete set which has been shown over recent years to carry a remarkable amount of structure. In this project, the PI will use the techniques of geometric group theory (groups acting on metric spaces) to study this set in two scenarios. The first is when the target group is a hyperbolic group. In this case the focus is algorithmic, and we search for algorithms to describe the set of homomorphisms explicitly. This will lead on to an algorithmic study of the logic of hyperbolic groups. The second scenario is when the target group is the mapping class group of S, an orientable surface of finite-type.If B is a nice space, then the set of (isomorphism classes of) S-bundles over B is naturally parametrised by (conjugacy classes of) homomorphisms from the fundamental group of B to the mapping class group of S. Thus the study of this set of homomorphisms is of fundamental interest to the study of surface bundles, a subject of broad interest throughout mathematics.Groups arise as the set of symmetries of a mathematical object. Thinking of a symmetry as a physical motion, one can `do nothing', `perform one symmetry and then another', and `undo a symmetry'. This gives algebraic structure to the set of symmetries, and abstracting this idea gives a group. This project will study a group G using the geometric properties of a space of which G is the set of symmetries. It will tackle questions about algorithms to solve equations over groups, which take their motivation from theoretical computer science and logic.A second focus is the study of surface bundles using group theory. A surface bundle is a space which locally looks like a product of two lower dimensional spaces (one of which is two dimensional), and they arise throughout pure mathematics.