Rögnvaldur G. Möller

Ends of graphs

It is shown how questions about ends of locally finite graphs can be reduced to questions about trees. Several applications are given; for example, locally finite connected graphs with infinitely many ends and automorphism groups that act transitively on the ends are classified.

Notes on infinite permutation groups

Some group theory.- Groups acting on sets.- Transitivity.- Primitivity.- Suborbits and orbitals.- More about symmetric groups.- Linear groups.- Wreath products.- Rational numbers.- Jordan groups.- Examples of Jordan groups.- Relations related to betweenness.- Classification theorems.- Homogeneous structures.- The Hrushovski construction.- Applications and open questions.

Descendants in highly arc transitive digraphs

A digraph is said to be highly arc transitive if its automorphism group acts transitively on the set of s-arcs for all s ≥ 0. The set of descendants of a directed line is defined as the set of all vertices that can be reached by a directed path from some vertex in the line. The structure of the subdigraph in a locally finite highly arc transitive digraph spanned by the set of descendants of a line is described and this knowledge is used to answer a question of Cameron, Praeger and Wormald. In addition another question of Cameron, Praeger and Wormald is settled.

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.

Highly arc transitive digraphs: reachability, topological groups

Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime in- or out-degree.Two different aspects of the connection between highly arc transitive digraphs and the theory of totally disconnected locally compact groups are also considered.

Locally-finite connected-homogeneous digraphs

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.

Accessibility and Ends of Graphs

We prove that a locally finite inaccessible graph with a transitive automorphism group always has uncountably many thick ends. Combined with a result of Thomassen and Woess this shows that a connected locally finite transitive graph is inaccessible if and only if it has uncountably many thick ends.

Distance-transitivity in infinite graphs

Abstract We give a positive answer to a question of Thomassen and Woess; we prove that for an infinite locally finite connected graph with more than one end 2-distance-transitivity implies distance-transitivity.

Digraphical Regular Representations of Infinite Finitely Generated Groups

A directed Cayley graphXis called a digraphical regular representation (DRR) of a groupGif the automorphism group ofXacts regularly onX. LetSbe a finite generating set of the infinite cyclic groupZ. We show that a directed Cayley graphX(Z,S) is aDRRofZif and only ifS?S?1. IfX(Z,S) is not aDRRwe show thatAut(X(Z,S)) =D∞. As a general result we prove that a Cayley graphXof a finitely generated torsion-free nilpotent groupNis aDRRif and only if no non-trivial automorphism ofNof finite order leaves the generating set invariant.

Topological groups, automorphisms of infinite graphs and a theorem of Trofimov

Abstract We give a short proof of a theorem of Trofimov, using the theory of topological groups. An automorphism g of a graph is bounded if there is a number M such that the distance between a vertex v and gv is less than M for all vertices v in the graph. Trofimovs theorem is a characterization of those locally finite infinite graphs that admit a transitive group of bounded automorphisims.