TY - JOUR
TI - Metrically homogeneous graphs
DO - https://doi.org/doi:10.7282/t3-exe2-jg62
PY - 2019
AB - We investigate the properties of graphs which are homogeneous in the sense of Fraisse when considered as metric spaces with the graph metric (metrically homogeneous graphs), and particularly the metrically homogeneous graphs of generic type constructed by Cherlin.
We first consider the properties of the associated automorphism groups, viewed as topological groups. For a large class of metrically homogeneous graphs of generic type, we show that the automorphism groups have ample generics, and therefore have a variety of topological properties such as the small index property and automatic continuity. We also show that the automorphism groups of the generic expansions of these graphs by linear orders are extremely amenable, and describe the universal minimal glow for the full automorphism group. Using standard model theoretic and descriptive set theoretic methods, these results are derived from the study of combinatorial properties of the associated classes of finite partial substructures.
Turning to more algebraic questions, we determine the twisted automorphism groups of metrically homogeneous graphs, and more generally the twisted isomorphisms between such graphs; these are isomorphisms up to a permutation of the natural language. Returning to the standard automorphism group, we then study the algebra of the associated age in the sense of Peter Cameron, showing that in most cases this algebra is a polynomial algebra. For this, we apply a criterion of Cameron based on a unique decomposition theorem.
KW - Mathematics
KW - Automorphisms
KW - Spaces of homogeneous type
LA - eng
ER -