arXiv subjects

Rögnvaldur G. Möller

Publications and source records attributed to Rögnvaldur G. Möller.

10 recordsLinked to original sources

arXiv2026

Graphs of group actions and group actions on trees

Bass-Serre theory provides a powerful framework for studying group actions on trees. While extremely effective for structural questions in group theory, it is less suited to the systematic construction of group actions with prescribed local behaviour. Motivated by local-to-global constructions such as the Burger-Mozes universal groups and local action diagrams, we develop an analogue of Bass-Serre theory for group actions. The central object of study in our are graphs of group actions, combinatorial structures similar to graphs of groups from Bass-Serre theory, encoding compatible local permutation actions on a base graph. From these we can construct groups which act on tree-like graphs called scaffoldings and hence also on trees. We prove uniqueness and universality results for the resulting groups and show that our framework unifies and generalises (among other known constructions) both graphs of groups and local action diagrams. Remarkably, we are able to encapsulate the full generality of the former while still allowing for efficient construction of groups with certain local properties like in the latter.

math.GR↗

arXiv2025

Periodic colorings and orientations in infinite graphs

We study the existence of periodic colorings and orientations in locally finite graphs. A coloring or orientation of a graph $G$ is periodic if the resulting colored or oriented graph is quasi-transitive, meaning that $V(G)$ has finitely many orbits under the action of the group of automorphisms of $G$ preserving the coloring or the orientation. When such a periodic coloring or orientation of $G$ exists, $G$ itself must be quasi-transitive and it is natural to investigate when quasi-transitive graphs have such periodic colorings or orientations. We provide examples of Cayley graphs with no periodic orientation or non-trivial coloring, and examples of quasi-transitive graphs of treewidth 2 without periodic orientation or proper coloring. On the other hand we show that every quasi-transitive graph $G$ of bounded pathwidth has a periodic proper coloring with $χ(G)$ colors and a periodic orientation. We relate these problems with techniques and questions from symbolic dynamics and distributed computing and conclude with a number of open problems.

math.CO↗

arXiv2021

Cayley--Abels graphs and invariants of totally disconnected, locally compact groups

A connected, locally finite graph $Γ$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $Γ$. Define the minimal degree of $G$ as the minimal degree of a Cayley--Abels graph of $G$. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_d$ denotes the $d$-regular tree, then the minimal degree of ${\rm Aut}(T_d)$ is $d$ for all $d\geq 2$.

math.GR↗

arXiv2021

Trivalent vertex-transitive graphs with infinite vertex-stabilizers

We study groups acting vertex-transitively on connected, trivalent graphs such that stabilizers of vertices are infinite. If the action is edge-transitive, we prove that the graph has to be a tree. We analyze the case where the action is not edge-transitive and fully classify the possible $2$-ended graphs. We draw connections to Willis' scale function and re-prove a result by Trofimov.

math.CO↗

arXiv2018

Infinite arc-transitive and highly-arc-transitive digraphs

A detailed description of the structure of two-ended arc-transitive digraphs is given. It is also shown that several sets of conditions, involving such concepts as Property Z, local quasi-primitivity and prime out-valency, imply that an arc-transitive digraph must be highly-arc-transitive. These are then applied to give a complete classification of two-ended highly-arc-transitive digraphs with prime in- and out-valencies.

math.CO↗

arXiv2018

Sharply $k$-arc-transitive-digraphs: finite and infinite examples

A general method for constructing sharply $k$-arc-transitive digraphs, i.e. digraphs that are $k$-arc-transitive but not $(k+1)$-arc-transitive, is presented. Using our method it is possible to construct both finite and infinite examples. The infinite examples can have one, two or infinitely many ends. Among the one-ended examples there are also digraphs that have polynomial growth.

math.CO↗

arXiv2018

On tree-decompositions of one-ended graphs

A graph is one-ended if it contains a ray (a one way infinite path) and whenever we remove a finite number of vertices from the graph then what remains has only one component which contains rays. A vertex $v$ {\em dominates} a ray in the end if there are infinitely many paths connecting $v$ to the ray such that any two of these paths have only the vertex $v$ in common. We prove that if a one-ended graph contains no ray which is dominated by a vertex and no infinite family of pairwise disjoint rays, then it has a tree-decomposition such that the decomposition tree is one-ended and the tree-decomposition is invariant under the group of automorphisms. This can be applied to prove a conjecture of Halin from 2000 that the automorphism group of such a graph cannot be countably infinite and solves a recent problem of Boutin and Imrich. Furthermore, it implies that every transitive one-ended graph contains an infinite family of pairwise disjoint rays.

math.CO↗

arXiv2015

Local finiteness, distinguishing numbers and Tucker's conjecture

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices.

math.CO↗

arXiv2012

Normal subgroups of groups acting on trees and automorphism groups of graphs

Let $T$ be a tree and $e$ an edge in $T$. If $C$ is a component of $T\setminus e$ and both $C$ and its complement are infinite we say that $C$ is a half-tree. The main result of this paper is that if $G$ is a closed subgroup of the automorphism group of $T$ and $G$ leaves no non-trivial subtree invariant and fixes no end of $T$ then the subgroup generated by the pointwise stabilizers of half-trees is topologically simple. This result is used to derive analogues of recent results of Caprace and De Medts (2011) and it is also applied in the study of the full automorphism group of a locally finite primitive graph with infinitely many ends.

math.GR↗

arXiv2009

Hyperbolic groups have flat-rank at most 1

The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flat-rank at most 1. It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank at most 1.

math.GR↗