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This thesis describes a number of algorithms and properties relating to Gromov’s\ud word-hyperbolic groups. A fuller outline of the thesis is given, and a number of\ud basic concepts relating to metric spaces, hyperbolicity and automaticity are first\ud briefly detailed in Chapter 1. Chapter 2 then details a solution to the conjugacy\ud problem for lists of elements in a word-hyperbolic group which can be run in linear\ud time; this is an improvement on a quadratic time algorithm for lists which contain\ud an infinite order element. Chapter 3 provides a number of further results and\ud algorithms which build upon this result to efficiently solve problems relating to quasiconvex\ud subgroups of word-hyperbolic groups – specifically, the problem of testing\ud if an element conjugates into a quasiconvex subgroup, and testing equality of double\ud cosets. In Chapter 4, a number of properties of certain coset Cayley graphs are\ud studied, in particular showing that graph morphisms which preserve edge labels and\ud directions and map a quasiconvex subset to a single point also preserve a variety of\ud other properties, for instance hyperbolicity. Finally, Chapter 5 gives a proof that all\ud word-hyperbolic groups are 14-hyperbolic with respect to some generating set

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