Infinite primitive groups Academic Article uri icon


  • We give a complete characterization of finitely generated primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as finitely generated subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings. In particular, we settle a conjecture of G. Higman and BH Neumann on the Frattini group of an amalgamated product, and prove that the Frattini subgroup of any non-elementary convergence group is finite, generalizing a theorem of I. Kapovich.

publication date

  • January 8, 2006