Homomorphisms to hyperbolic and mapping class groups

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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.