Countable Primitive Groups
We give a complete characterization of countable 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 subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product.
Keywords and phrases:Permutation groups primitive actions maximal subgroups linear groups mapping class groups hyperbolic groups frattini subgroups
AMS Mathematics Subject Classification:Primary 20B15 Secondary 20B07, 20E28
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