Ashot Minasyan

Invariable generation for infinite groups

A group G is said to be invariably generated (IG) if it cannot be covered by conjugates of a proper subgroup. This is equivalent to the statement that for every transitive action of G on a set with at least 2 elements, there is element of G acting without fixed points. The fact that this holds for every finite group was proved by Camille Jordan in 1872, and the property was later rediscovered and studied independently by Wiegold, Dixon and Serre.

Every IG group possesses an invariable generating subset S, such that if one replaces elements of S by arbitrary conjugates, the resulting subset still generates the group. Recently Kantor, Lubotzky and Shalev introduced and studied the class of finitely invariably generating groups (FIG), which have finite invariable generating subsets.

Basic examples of IG groups are abelian groups, which are FIG iff they are finitely generated. It is known that IG and FIG groups are closed under extensions, hence virtually solvable groups are IG and virtually polycyclic groups are FIG.

In 1977 Wiegold asked if the property IG is invariant under commensurability, i.e., whether it is stable under taking finite index subgroups. A similar question for FIG was asked by Kantor-Lubotzky-Shalev in 2015. In my talk I will discuss a construction of a FIG group that has a non-IG index 2 subgroup, which answers both of the above questions negatively.