On graphs and valuations

Abstract

In the last two decades new techniques emerged to construct valuations on an infinite division ring $D,$ given a normal subgroup $N\subseteq D$ of finite index. These techniques were based on the commuting graph of $D^{\times}/N$ in the case where $D$ is non-commutative, and on the Milnor K-graph on $D^{\times}/N,$ in the case where $D$ is commutative. In this paper we unify these two approaches and consider V-graphs on $D^{\times}/N$ and how they lead to valuations. We furthermore generalize previous results to situations of finitely many valuations.