Geometric and Functional Analysis

, Volume 17, Issue 5, pp 1479–1523

Countable Primitive Groups


DOI: 10.1007/s00039-007-0630-y

Cite this article as:
Gelander, T. & Glasner, Y. GAFA Geom. funct. anal. (2008) 17: 1479. doi:10.1007/s00039-007-0630-y


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 groupsprimitive actionsmaximal subgroupslinear groupsmapping class groupshyperbolic groupsfrattini subgroups

AMS Mathematics Subject Classification:

Primary 20B15Secondary 20B07, 20E28

Copyright information

© Birkhaeuser 2007

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsBen Gurion University of the NegevBe’er ShevaIsrael