RWHG

Random walks on hyperbolic groups

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 Obiettivo del progetto (Objective)

'The project lies at the confluent of different mathematical fields: probability theory, algebra, and geometry. There is a contribution of A. V. Vershik in a special Springer volume about the future of mathematics in the 21st century that points out the prospects and challenges that are comprised in the interplay of probability theory and algebra. Here random walk theory, a branch of probability theory, plays a mayor role. There are two points of view to look at the relation between probability theory, algebra, and geometry. The probabilistic viewpoint concerns all questions regarding the impact of the underlying structure on the behavior of the corresponding random walk. Typically one is interested in transience/recurrence, spectral radius, rate of escape, and central limit theorems. On the other hand, random walks are a useful tool to describe the structure that underlies the random walk. In particular, algebraic and geometric properties can be classified due to the behaviour of the corresponding random walks. The project falls exactly into this topic: we will study random walks on hyperbolic groups. The objectives are to prove a central limit theorem for random walks on hyperbolic groups and provide geometric interpretations of the asymptotic variance. This will arise from a geometric perspective in the flavour of the interpretation for the rate of escape in terms of entropy and requires deeper knowledge of hyperbolic geometry together with inspiration and new ideas. The project will settle the ground for future collaboration, not only between France and Germany but also on an European level, since the host institute and the applicant have strong European contacts. Furthermore, the project can be seen as a continuation and complement of the existing Marie Curie contract ``European Training Courses and Conferences in Group Theory''.'

Introduzione (Teaser)

An EU-funded project on pure mathematics sought out links between different areas of mathematics. Such discoveries could provide new insights that could be adapted to resolve complex problems in diverse fields ranging from physics to economics.

Descrizione progetto (Article)

The 'Random walks on hyperbolic groups' (RWHG) project, funded by the EU, explored the area where three different fields of mathematics converged: probability theory, algebra and geometry. Although these fields may seem quite isolated, random walk theory has a major role to play in the area of their convergence. It is a branch of probability theory that mathematically represents a path comprising a succession of random steps.

In general, there are two points of view with regard to the relationship between probability theory and algebra and geometry. The probabilistic viewpoint addresses questions on the impact of the underlying structure on the behaviour of the corresponding random walk. Random walks, however, describes the dynamics or behaviours of the structure-of-interest. In particular, algebraic and geometric properties can be classified based on the behaviour of their corresponding random walks.

A specific aim of the RWHG project was to find a classification of algebraic structures that is defined by an analogue of the central limit theorems. The project set out to give a detailed and exhaustive list of all the abstract objects that satisfy their definition.

The RWHG project thus realised the first step towards this ambitious objective. A central limit theorem was successfully proved for co-compact fuchsian hyperbolic groups. The proof proposed by the project could be applied to large classes of mathematical objects or groups.

Unfortunately, the project ended prior to maturity, but it is hoped that further work will be done in this very complex field of mathematics.