On generalised Farey graphs and applications to the curve complex

In the first part of the thesis, we introduce a family of simplicial complexes called tree complexes, which generalise the well-known Farey graph. We study numerous aspects of tree complexes. Firstly we show for a given dimension n, the tree complex K(n) is simplicially rigid. We then study the geodesics between a pair of given vertices x and y, giving a bound in terms of the distance between the vertices, and showing that there always exist a pair of vertices at a given distance which attains this bound. When n = 2, this bound is the ith Fibonacci number, where i is the distance between the two vertices. We next study the automorphism group of a tree complex, showing that it splits as a semi-direct product. Finally we study the coarse geometry of a tree complex, showing in particular that for n > 2 each tree complex is quasi-isometric to the simplicial tree T [infinity]. In the second part of the thesis, we study the curve complex of the five-holed sphere, C(S0,5), via subsurface projections to the four-holed sphere. We show that geodesically embedded pentagons, hexagons and heptagons are unique, up to the action of the mapping class group. We conjecture firstly that there are no larger geodesically embedded cycles in C(S0,5), and secondly that these methods might be used in a greatly simplified proof of the hyperbolicity of C(S0,5)