In a joint work with Tsachik Gelander. We study finitely
generated groups in terms of their actions on sets.
Elaborating on methods developed by Margulis and Soifer we investigate
finitely generated groups that admit a faithful primitive action on a set.
We are able to give a complete classification of such groups in various
geometric settings including linear groups, subgroups of hyperbolic
groups, subgroups of mapping class groups and more.
Primitive actions can be thought of as the “irreducible representations”
in the theory of permutation representations. By definition these are the
actions in which no non-trivial equivalence relation is preserved.
The fact that a group admits a primitive permutation representation
carries some significant group theoretic information about the group. In
the setting of linear groups, every finitely generated group that has a
simple Z-closure also admits a faithful primitive permutation
representation. An necessary and sufficient condition for admitting such
an action is slightly more technical.
As an application of our classification of primitive groups we prove a
conjecture of Higman and Neumann on the Frattini subgroup of an
amalgamated free product.