Cyclic orders and graphs of groups (Q6657209)

A well-known result of \textit{A. A. Vinogradov} [Mat. Sb., N. Ser. 25(67), 163--168 (1949; Zbl 0038.15904)], asserts that free groups, and more generally, free products of left-orderable groups are left-orderable. \textit{W. Dicks} and \textit{Z. Šunić}, in [Can. Math. Bull. 63, No. 2, 335--347 (2020; Zbl 1481.06039)], gave an elegant way of totally ordering the vertex set of a directed tree and they applied this to give a simple proof of Vinogradov's result. Cyclically ordered group are a bit more general than ordered groups (see [\textit{É. Ghys}, Enseign. Math., II. Sér. 47, No. 3--4, 329--407 (2001; Zbl 1044.37033)] and [\textit{D. Calegari}, Geom. Topol. Monogr. 7, 431--491 (2004; Zbl 1181.57022)] for surveys on cyclically ordered groups).\N\NThe paper under review is devoted to describe a cyclically ordered counterpart of Vinogradov's result. In particular the authors prove Theorem 1.2: Let \(G\) split as a graph of groups with left-cyclically ordered vertex groups and convex left-ordered edge groups. Then \(G\) is left-cyclically ordered in a manner compatible with its vertex and edge groups.