Abelian structure in approximate groups and Alon's conjecture on Ramsey Cayley graphs

TL;DR

This paper extends results on the structure of approximate groups in solvable and other groups, proving new bounds and applying these to problems in graph theory and additive combinatorics.

Contribution

It establishes polynomial bounds for approximate subgroups in linear groups and verifies Alon's conjecture for almost all finite groups, advancing understanding of group structure and graph properties.

Findings

Existence of large abelian intersections in approximate groups within solvable groups.

Verification of Alon's conjecture for almost all finite groups.

Polynomial bounds for approximate subgroups of linear groups.

Abstract

A result of Pyber states that every finite group $G$ contains an abelian subgroup whose order is quasi-polynomially large in $\lvert G\rvert$. We prove a similar result for $K$-approximate subgroups of solvable groups under only modest restrictions on $K$. We show that, if $A$ is a finite $K$-approximate group contained in some solvable group, then some abelian group intersects $A^4$ in at least $\exp(\Omega(\log^{1/6}\lvert A\rvert/\log 2K))$ elements. We also prove a similar result for approximate subgroups of finite groups with no large alternating subquotients. Along the way, we obtain polynomial (instead of quasi-polynomial) bounds for the same statement of approximate subgroups of linear groups. We give two applications. Firstly, we consider the conjecture of Alon that every finite group $G$ admits a Cayley graph with clique number and independence number $O(\log\lvert G\rvert)$. Conlon, Fox, Pham, and Yepremyan have recently proven that, for almost all positive integers $N$, every abelian group of order $N$ satisfies Alon's conjecture. Extending their result, we verify Alon's conjecture for all (not necessarily abelian) groups of almost all orders. Secondly, we prove a "local" version of Roth's theorem in (many) non-abelian settings with quasi-polynomial bounds, using the recent breakthroughs of Kelley and Meka on Roth's theorem and of Jaber, Liu, Lovett, Ostuni, and Sawhney on the corners problem.