Let \(D\) be an infinite division ring (associative), \(D^*\) be its multiplicative group, \(N\) a proper normal finite index subgroup of \(D^*\) containing \(-1\). The paper under review considers the problem of finding conditions that guarantee the existence of a valuation on \(D\) with nice properties. Several methods to detect valuations are known in case \(D\) is commutative, notably the rigidity method based on the Milnor \(K\)-groups of \(D\) relative to the subgroup \(N\leq D^*\) [see, e.g., I. Efrat, Isr. J. Math. 172, 75-92 (2009; Zbl 1197.19001)]. When \(D\) is noncommutative, new techniques to solve this problem emerged in the last two decades, and they were based on the commuting graph of \(D^*/N\) [see Y. Segev, Ann. Math. (2) 149, No. 1, 219-251 (1999; Zbl 0935.20009); A. S. Rapinchuk and Y. Segev, Invent. Math. 144, No. 3, 571-607 (2001; Zbl 0999.16016); and A. S. Rapinchuk et al., J. Am. Math. Soc. 15, No. 4, 929-978 (2002; Zbl 1008.16018)].
The paper under review unifies these two approaches, by considering valuation graphs (\(V\)-graphs), say \(\Delta\), on \(D^*/N\) and how they lead to valuations. Using the axioms of a \(V\)-graph in conjunction with additional hypotheses, such as assumptions on its diameter \(\text{diam}(\Delta)\), the authors produce a surjective group homomorphism \(\varphi\colon N\to\Gamma\) to a partially ordered group \(\Gamma\). They show that \(\varphi\) possesses certain special properties making it what is called in the paper a strongly levelled map, in case \(\text{diam}(\Delta)\geq 3\), a strong valuation-like map, when \(\text{diam}(\Delta)\geq 4\), and a strong valuation-like map of \(s\)-level zero, if \(\text{diam}(\Delta)\geq 5\).
Secondly, it is proved that if \(D\) is commutative or is a finite-dimensional algebra over a field \(E\) of finite transcendence degree over its prime subfield, and if \(D^*/N\) supports a \(V\)-graph \(\Delta'\) of diameter \(\geq 4\), then there exists a (nontrivial) valuation \(v\) on \(D\), such that \(N\) is open in \(D^*\) with respect to the topology defined by \(v\).
The third main result of the reviewed paper concerns the case where \([D:E]<\infty\), \(E\) is infinite and satisfies the above-noted conditions, the centre \(Z(D)\) is a separable extension of \(E\), the image of \(N\cap E^*\) under \(\varphi\) is an ordered subgroup of \(\Gamma\), and \(\text{diam}(\Delta')\geq 4\). It shows that then \(E\) has a real-valued valuation \(v\), such that \(N\cap E^*\) is open in the \(v\)-adic topology on \(k\), there is a nonempty finite set \(T\) of valuations of \(Z(D)\) extending \(v\), such that each \(w\in T\) is uniquely extendable to a valuation \(w'\) on \(D\), and \(N\) is open in the \(T'\)-adic topology of \(D^*\), where \(T'=\{w':w\in T\}\).