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Random walks on hyperbolic groups

Final Report Summary - RWHG (Random walks on hyperbolic groups)

The project lay at the confluent of the following 'different' fields of mathematics that may appear quite isolated at the first glance: probability theory, algebra, and geometry. Here random walk theory, a branch of probability theory, plays a major role. In general there are two points of view to look at the relation between probability theory on the one side and algebra and geometry on the other side. The probabilistic viewpoint concerns all questions regarding the impact of the underlying structure on the behaviour of the corresponding random walk. Typically one is interested in transience / recurrence, spectral radius, asymptotic behaviour of the transition probabilities, rate of escape, central limit theorems, harmonic functions and convergence to a boundary at infinity. On the other hand, random walks are a useful tool to describe the structure that underlies the dynamic. In particular, algebraic and geometric properties can be classified due to the behaviour of the corresponding random walks. One specific aim of the project was to find a classification on which algebraic structures an analogue of the central limit theorems holds true. A first step towards this ambitious objective was to consider hyperbolic groups where we successfully proved a central limit theorem for the important subclass of co-compact Fuchsian groups. Even better, our proof strategy seems to apply for large classes of groups that fulfil two main assumptions that we conjecture to hold for large classes of finitely generated groups.

The project ended prior to maturity since the host institute offered Sebastian Müller an assistant professorship. This situation is best possible in order to pursue the proposed research topics and underlines the good collaboration of the researchers involved.