Mutations of quivers were introduced by Fomin and Zelevinsky in the beginning of 2000’s in the context of cluster algebras. Since then, mutations appear (sometimes completely unexpectedly) in various domains of mathematics and physics. Using mutations of quivers, Barot and Marsh constructed a series of presentations of finite Coxeter groups as quotients of infinite Coxeter groups. We will discuss a generalization of this construction leading to a new invariant of bordered marked surfaces, and a geometric interpretation: it occurs that presentations constructed by Barot and Marsh give rise to a construction of geometric manifolds with large symmetry groups, in particular to some hyperbolic manifolds of small volume with proper actions of Coxeter groups. This work is joint with Pavel Tumarkin.