Self-similar groups and algebras

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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.