Automorphism groups of symmetric graphs of valency 3 (Q1823961)

In the present article some investigations of \textit{D. Z. Djoković} and \textit{G. L. Miller} [J. Comb. Theory, Ser. B 29, 195-230 (1980; Zbl 0385.05040)] and \textit{D. M. Goldschmidt} [Ann. Math., II. Ser. 111, 377- 406 (1980; Zbl 0475.05043)] are continued. It was shown in the cited papers that there are only seven groups G such that G is isomorphic to an automorphism group of the infinite cubic tree such that it acts symmetrically and each vertex has a finite stabilizer. The first part of this work deals with the precise description of the seven groups in terms of defining relations between words over a three- element generating system. Some particular relationships among the seven groups are also stated. A natural bijection between the vertices of the infinite cubic tree and certain words (called tree words) over two generators is established. In the article of Djoković and Miller (loc. cit.) seven open problems have been raised. Now the first problem is solved by a construction, and affirmative answers (obtained by examples) are given to two other questions (second and third problems).