Cluster algebras, Coxeter groups and hyperbolic manifolds

Coxeter群在数学中表现为许多对象的对称群，特别是正多面体(古希腊人已经知道的五个柏拉图立体)的对称群，以及在艺术和现实生活中使用的许多瓷砖的对称群。簇代数是Fomin和Zlevinsky在2002年提出的一个非常新的概念。此后不久，簇代数被证明与数学中的许多其他领域有关，如多面体的组合学、表示论、泊松几何、TeichMuller理论、可积系统等。这些联系聚集了来自数学和数学物理许多不同分支的研究人员，导致了簇代数理论和相关领域的惊人快速发展。自从簇代数被引入以来，人们就知道一些簇代数与一些Coxeter群相连。更确切地说，有限簇代数(唯一有限生成的)是由有限结晶学Coxeter群列举的。这个建议的第一个目的是将这种对应关系进一步推广到下一个复杂的簇代数类(称为有限突变类型的簇代数，包括从三角化边界曲面产生的一大类簇代数)。这类代数应该对应于Coxeter群的某些商。对于由曲面产生的代数，所构造的群中的关系应该对应于曲面上的某些路径，一旦构造了代数与群之间的对应，就会产生许多自然问题。特别地，有趣的是，一些被构造为Coxeter群的商的群本身是否也是Coxeter群？如果簇代数不是有限的，那么所构造的群能是有限的吗？不同的代数会导致不同的群吗？在曲面代数的情况下，得到的群如何连接到初始曲面？作为理论的应用之一，我们将构造具有大对称群的有限体积双曲流形。

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.