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Beschreibung:
This thesis deals with various topics concerning infinite graphs and finitely generated infinite groups using many ideas from topological infinite graph theory and geometric group theory. In Chapter 3, we extend algebraic flow theory of finite graphs to infinite graphs with ends via abelian Hausdorff topological groups. This is achieved by developing a new compactness method for arbitrary (not necessarily locally finite) infinite graphs. In Chapter 4 we prove some sufficient conditions on finitely generated groups in order to force the resulting Cayley graphs to have Hamilton circles. We find Hamilton circles by decomposing Cayley graphs into finite cycles, infinite circles and double rays and then joining them together via some intermediate paths. In Chapter 5 we continue our study of Hamilton circles of Cayley graphs of finitely generated infinite groups in particular, two ended group or context-free group. We focus on finding generating sets for a given group of this type such that the respective Cayley graphs contains Hamilton circles. In other words, by choosing a large enough generating set of a given such group, we ensure that the Cayley graph of the group with respect to that generating set contains a Hamilton circle. Furthermore, we determine the minimum possible size of such a generating set for a given two-ended group or context-free group. Chapter 6 deals with the structure of 2-ended graphs and 2-ended groups. We lift some standard characterisation of 2-ended groups to 2-ended quasi-transitive graphs without dominated ends. In Chapter 7, we study tree-decompositions of locally finite graphs with a certain amount of symmetry. We find specific tree-decompositions of a given graph which are compatible with the action of a group on the graph. Also, we find a graph-theoretical version of Stallings' theorem for locally finite quasi-transitive graphs. In the final chapter, we discuss some applications of Chapter 7. For example, we show that the graph-theoretical version of Stallings' theorem leads to a new ...