Cluster algebras, Coxeter groups and hyperbolic manifolds

点击翻译按钮获取中文摘要

Coxeter groups appear in mathematics as symmetry groups of many objects, in particular, symmetry groups of regular polyhedra (the five Platonic solids known already to ancient Greeks) as well as symmetry groups of many tilings used both in art and real life. Cluster algebras is a very recent notion introduced by Fomin and Zelevinsky in 2002. Soon after that, it turned out that cluster algebras are connected to many other fields in mathematics, such as combinatorics of polytopes, representation theory, Poisson geometry, Teichmuller theory, integrable systems. These connections brought together researchers from many different branches of mathematics and mathematical physics, which induced amazingly rapid growth both of the theory of cluster algebras and of related fields.It was known since introduction of cluster algebras that some cluster algebras are connected to some Coxeter groups. More precisely, finite cluster algebras (the only ones which are finitely generated) are enumerated by finite crystallographic Coxeter groups. The first aim of this proposal is to push this correspondence further to the next complexity class of cluster algebras (called cluster algebras of finite mutation type and including a large class of cluster algebras arising from triangulated boardered surfaces). Algebras from this class should correspond to certain quotients of Coxeter group. For algebras arising from surfaces, the relations in the constructed group should correspond to certain paths on the surface.There are many natural questions arising once the correspondence between algebras and groups constructed. In particular, it is interesting to know if some of the groups constructed as quotients of Coxeter groups are Coxeter groups themselves? Can the constructed group be finite if the cluster algebra is not a finite one? Do different algebras induce different groups? How the obtained group is connected to the initial surface, in the case of a surface algebra?As one of the applications of the theory, we will construct finite volume hyperbolic manifolds with large symmetry groups.