Small doubling in right-ordered groups

Abstract: Freiman conjectured that if $S$ is a finite subset of a torsion-free group $G$ with $k\geq 3$ elements and $|S^{2}|\leq 3k-4,$ then $S$ is a subset of a small geometric progression of length at most $2k-3$. In 2014, Freiman et al. settled this conjecture when $S$ is a finite subset of an ordered group. In this talk, we study this problem in the broader framework of right-ordered groups. Under suitable structural conditions on the subset $S$, we discuss results that extend aspects of Freiman's conjecture to this setting. We further focus on the right-ordered Baumslag--Solitar group $$\text{BS}(1,q) = \langle a, b \mid ab = b^q a \rangle, \quad q \in \mathbb{Z}.$$ We show that for $q \neq -1$, if $S$ is a finite subset of $\text{BS}(1,q)$ with the identity element as its minimum and satisfying $|S^2| \leq 3|S| - 4$, then the subgroup generated by $S$ is abelian. This is joint work with Mohan and B. R. Shankar. The results are based on our recent paper: link.springer.com/article/10.1007/s00025-025-02576-2.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)