Subgroups of profinite groups acting on trees

P. A. Zalesskii ; O. V. Mel'nikov

Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 405-424

Résumé

The fundamental groups $\Pi_1(\mathscr G,\Gamma)$ of finite graphs of profinite groups are studied here; the definitions are similar to those of the analogous constructions in Bass-Serre theory. Results are obtained on the following: the disposition of finite and finite normal subgroups of $\Pi_1(\mathscr G,\Gamma)$; intersections of conjugates of vertex-groups; and normalizers of vertex-groups. In the case where $\Pi_1(\mathscr G,\Gamma)$ is not an amalgamated free product $A*_NB$, where $[A:N]=2=[B:N]$, it is proved that a normal subgroup of $\Pi_1(\mathscr G,\Gamma)$ either lies in every edge-group or else has a nonabelian free profinite subgroup. Proofs are based on the examination of fixed elements of profinite trees with profinite groups acting on them. The definition of profinite tree is close to that in the article RZh.Mat., 1978, 11A232. Some of our results have been proved already, but only for free products of profinite groups. The methods were different from those in this paper (see RZh.Mat., 1979, 9A180 and 1985, 8A232). Bibliography: 16 titles.