Bernhard Krön

End compactifications in non-locally-finite graphs

There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and give a proof of the compactness of the end-topology that is constructed by the principle of removing finite sets of edges. For the latter case there exists already a proof in \cite{cartwright93martin}, which only works on graphs with countably infinite vertex sets and in contrast to which we do not use the Theorem of Tychonoff. We also construct a new topology of ends that arises from the principle of removing sets of vertices with finite diameter and give applications that underline the advantages of this new definition.

Quasi-isometries between graphs and trees

Criteria for quasi-isometry between trees and general graphs as well as for quasi-isometries between metrically almost transitive graphs and trees are found. Thereby we use different concepts of thickness for graphs, ends and end spaces. A metrically almost transitive graph is quasi-isometric to a tree if and only if it has only thin metric ends (in the sense of Definition 3.6). If a graph is quasi-isometric to a tree then there is a one-to-one correspondence between the metric ends and those d-fibers which contain a quasi-geodesic. The graphs considered in this paper are not necessarily locally finite.

Quasi-isometries between non-locally-finite graphs and structure trees

We prove several criteria for quasi-isometry between non-locally-finite graphs and their structure trees. Results ofMöller in [11] for locally finite and transitive graphs are generalized. We also give a criterion in terms of correspondence between the ends of the graph and the ends of the structure tree.

LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS

We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

On stallings' unique factorisation groups

Let be a group and a symmetric generating set for . In (8), Stallings called a unique factorization group if each group element may be written in a unique way as a product a1 : : : am, where ai 2 for each i and aiai+1 62 ( f1g for each i < m. In this paper we give a complete combinatorial proof of a theorem, not explicitly stated in (8), characterizing all such pairs ( ; ). We also characterize the unique factorization pairs by a certain tree-like property of their Cayley graphs.