Homomorphisms to hyperbolic and mapping class groups

一对（离散）群之间的同态集合是近年来被证明具有大量结构的离散集合。在这个项目中，PI将使用几何群论（群作用于度量空间）的技术在两种情况下研究这个集合。第一种是当目标群体是双曲群体时。在这种情况下，重点是算法，我们寻找算法来显式地描述同态集合。这将导致对双曲群逻辑的算法研究。第二种情况是目标群为S的映射类群，S是有限型的可定向曲面。如果B是一个良好的空间，那么B上的s束的同构类集自然地被B的基本群到s的映射类群的同态类所参数化。因此，对这组同态集的研究对于曲面束的研究具有重要意义，而曲面束的研究是整个数学中广泛关注的课题。群产生于数学对象的对称集合。把对称看作是一种物理运动，人们可以“什么都不做”，可以“实现一个又一个对称”，也可以“撤销一个对称”。这就给出了对称集合的代数结构，抽象这个思想就得到了群。这个项目将使用空间的几何性质来研究群G，其中G是对称的集合。它将解决关于解决群体方程的算法问题，这些问题的动机来自理论计算机科学和逻辑。第二个重点是用群论研究表面束。表面束是一个局部看起来像两个低维空间（其中一个是二维空间）乘积的空间，它们在纯数学中出现。

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.