FWF - Hyperbolic Structures - Hyperbolic Structures in Stochastics, Graph Theory, and Topology (FWF P24028)

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 - Poincar 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. Continuation of the study of Brownian motion and harmonic functions on "treebolic space" and related horocyclic products. Random walks on Baumslag-Solitar groups - related with A. Study of stochastic dynamical systems of iterated Lipschitz mappings in the critical case via the hyperbolic extension introduced by Peign and Woess, and processes on hyperbolic boundaries. A continuation of the work begun by Georgakopoulos und Kolesko on Brownian motion and potential theory on metric graphs with finite total edge lengths. Characterisation of the planar groups, yielding structure tree splittings of these groups; extensions to the non-planar case. 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.