Self-similar groups and algebras

自相似或分形的物体在几何学中比比皆是，但它们对代数的影响相对较新。群、结合代数和李代数称为自相似的，如果它们配备了一个双集(分别是双模)，即一个集合(分别是。模块)，具有向左和向右的通勤动作，这是自由的向右集合(分别为模块)。重要的例子包括Grigorchuk的无限扭群和指数增长群等。一个动力系统可以方便地编码为一个自相似群，这产生了该动力系统的一个极强的代数不变量，以及动力学和代数之间的联系。建议集中在自相似代数对象的两个主要方面：(1)具有显著新特征的无限维代数的构造；(2)群论和全纯动力学之间的联系，特别是高维映射。

Self-similar, or fractal, objects abound in geometry, but their impact on algebra is relatively new. Groups, associative algebras, and Lie algebras are called self-similar if they are equipped with a biset (respectively bimodule), namely a set (resp. module) with commuting left and right actions, which is free qua right set (resp. module). Important examples include the infinite torsion groups, and groups of exponential growth, by Grigorchuk inter alia. A dynamical system may be conveniently encoded as a self-similar group; this yields an extremely potent algebraic invariant of that dynamical system, and a link between dynamics and algebra.The proposal will concentrate on two major aspects of self-similar algebraic objects: (1) the construction of infinite-dimensional algebras with striking new features; and (2) the links between group theory and holomorphic dynamics, with particular focus on higher-dimensional mappings.