  We construct finite volume hyperbolic manifolds with large symmetry groups.
The construction makes use of the presentations of finite Coxeter groups
provided by Barot and Marsh and involves mutations of quivers and diagrams
defined in the theory of cluster algebras. We generalize our construction by
assigning to every quiver or diagram of finite or affine type a CW-complex with
a proper action of a finite (or affine) Coxeter group. These CW-complexes
undergo mutations agreeing with mutations of quivers and diagrams. We also
generalize the construction to quivers and diagrams originating from
unpunctured surfaces and orbifolds.
