Abelian structure in arbitrary approximate groups

Establish that there exists an absolute constant c>0 such that every group G and every finite symmetric set A satisfying |A^3|≤K|A| contain a commuting set T⊂A^4 with |T|≥exp(Ω(log^c|A|/log 2K)), without assuming that G avoids alternating-group subquotients.

Background

The paper proves a quasi-polynomially large commuting subset inside A^4 for solvable groups and for finite groups with no subquotient isomorphic to a sufficiently large alternating group. The authors conjecture that the alternating-subquotient restriction is unnecessary, which would extend the principal abelian-structure theorem to arbitrary ambient groups.

The conjecture would remove all ambient-group restrictions from the paper’s non-abelian local Roth theorem and would strengthen its results on Ramsey Cayley graphs. In the special case K=1, it would also yield a weak form of Pyber’s theorem without relying on the classification of finite simple groups.