The cyclomatic number of connected graphs without solvable orbits (Q2906836)

The authors prove that there exists a connected multigraph \(G\) of cyclotomic number \(c\) without solvable orbits if and only if \(c \notin S = \{ 0, \dots ,5,7,9,12,13,14,17,18,23 \}\). The cyclotomic number of \(G\) is defined to be one minus the Euler characteristic of \(G\) and ``without solvable orbits'' means that \(\mathrm{Aut}(G)\) acts on each orbit via a non-solvable quotient. Furthermore, if \(g\notin S\) and \(K\) is a field complete relative to a discrete valuation, they show that there exists a smooth, projective, geometrically irreducible curve of genus \(g\) over \(K\) without solvable points over \(K\).