Boundaries of infinite graphs: ends, group actions and random walks

This research project is dedicated to three aspects of boundary theory in infinite graphs and their interplay. One emphasis are non-locally finite graphs and infinitely generated groups. 1. Structural graph theory. We want to discuss several connections between b- and d-fibres (in the sense of Jung) and metric ends. Work of Halin concerning periodic lines shall be generalized to the non-locally finite case. We are interested in topological properties of end compactifications. Creteria for quasi-isometries between graphs and trees are of pure graph theoretic nature, but they may have an impact on group theoretic questions. 2. Group theory. A finitely generated group is virtually free if and only if it has a finitely generated Cayley graph which is quasi-isometric to a tree (Gromov, Woess). We want to find generalization of this theorem for infinitely generated groups. 3. Random walks and potential theory. In how far can the Dirichlet problem be solved on the metric end boundary of graphs or on the end boundary of the countably infinite generated free group? We hope to find a construction for probabilistic ends such that we can describe the Martin boundary for certain cases of irreducible but not uniformly irreducible random walks on graphs. We also want to discuss the Martin boundary of a certain type of buildings.