Hyperbolic Structures in Stochastics, Graph Theory, and Topology

The hyperbolic structures of the project title are generally metric spaces which are hyperbolic in the sense of Gromov. This comprises infinite trees as well as the classical Bolyai - Lobachevski - Poincaré hyperbolic plane. More generally, the latter structures appear as sub-structures of other geometric objects such as metric as well as combinatorial graphs, horocyclic productsof trees and hyperbolic planes, and other related Riemannian complexes. In particular, the research focusses on Graphs which (with their discrete metric) are hyperbolic in the sense of Gromov; Trees as basic examples of hyperbolic graphs, also used for describing the structure of more complicated spaces via structure trees; Hyperbolic spaces or graphs that are constructed from more general metric spaces, with the latter as their boundary at infinity; Spaces (not necessarily only graphs) that are not hyperbolic themselves, but have hyperbolic "building blocks", such as the horocyclic products of trees and the upper half plane that were studied in the preceding project. The specific themes that we intend to investigate are the following. 1. Continuation of the study of Brownian motion and harmonic functions on "treebolic space" and related horocyclic products. 2. Random walks on Baumslag-Solitar groups - related with A. 3. Study of stochastic dynamical systems of iterated Lipschitz mappings in the critical case via the hyperbolic extension introduced by Peigné and Woess, and processes on hyperbolic boundaries. 4. A continuation of the work begun by Georgakopoulos und Kolesko on Brownian motion and potential theory on metric graphs with finite total edge lengths. 5. Characterisation of the planar groups, yielding structure tree splittings of these groups; extensions to the non- planar case. 6. Using augmented trees and Gromov-hyperbolicity to describe local connectedness in topological spaces. As it is typical for the research interests of W. Woess, these topics are brought together by the aim to do mathematical research at the meeting point of different fields, strongly featuring the interplay of structure theory with other topics. In comparison with previous projects, there is a partial shift from random walks to Brownian motion and also a new bias on topics between graph theory and topology. Of course, the study of random walk related subjects will remain present in different ways. This project and its employees (one PostDoc and one PhD student) are expected to interact strongly with the FWF- funded doctoral program (DK-plus) "Discrete Mathematics", of which Wolfgang Woess is the speaker.

This mathematical research project was concerned with questions which are of hyperbolic nature in different ways. This is to be understood in terms of geometry: paradigm is the hyperbolic plane, as it has been discovered in the 19th century by Lobatschevski, Bolyiai, as well as Gauss. It is negatively curved, and the classical parallel axiom is no more valid. It has led to other, here also discrete structures, that is, graphs and groups with hyperbolic properties. Graphs can be imagined as networks; here, they have infinitely many nodes which are connected by bonds (edges). The most typical graph-counterpart of hyperbolic plane is a regular tree. Groups can be understood as symmetry mappings of geometric structures. The topics of this research project concern on one hand the structure of graphs, groups and metric spaces with hyperbolic nature, and on the other hand random processes which take place in such structures: Markov processes, Random Walks, the associated analysis and potential theory. The topics of this project were laid out broadly and had, in particular, the aim to provide younger researchers with a fruitful environment and to further develop the vivid research group of Woess. In fact, there have been altogether 6 project collaborators, whose project activities have in large part lead them to subsequent academic positions. A large number of project publications was begun, carried out, resp. completed within this project. The main results concern the following topics: ? characterisation of the Poisson boundary of transient (hyperbolic) random walks on plane square tilings; ? algebraic and dynamical properties of automata groups, that is, symmetry groups of regular rooted trees which are generated by Mealy-automata; ? Brownian motion and harmonic functions on treebolic spaces (horocyclic products of the hyperbolic plane with regular trees, tree-like unfoldings of hyperbolic plane); ? characterisation of the Poisson boundary randes für random walks on non-amenable Baumslag-Solitar groups (symmetry groups of variants of the treebolic spaces); ? asymptotic analysis of isotropic Markovian random processes on ultra-metric spaces (the geometric boundaries of trees); ? investigation of transience and recurrence of reflected random walks in multi-dimensional space.