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Let $G$ be a countable abelian group. We study ergodic averages associatedwith configurations of the form $\{ag,bg,(a+b)g\}$ for some $a,b\in\mathbb{Z}$.Under some assumptions on $G$, we prove that the universal characteristicfactor for these averages is a factor of a $2$-step nilpotent homogeneousspace. As an application we derive a Khintchine type recurrence result. Inparticular, we prove that for every countable abelian group $G$, if$a,b\in\mathbb{Z}$ are such that $aG,bG,(b-a)G$ and $(a+b)G$ are of finiteindex in $G$, then for every $E\subset G$ and $\varepsilon>0$ the set $$\{g\inG : d(E\cap E-ag\cap E-bg\cap E-(a+b)g)\geq d(E)^4-\varepsilon\}$$ is syndetic.This generalizes previous results for $G=\mathbb{Z}$, $G=\mathbb{F}_p^\omega$and $G=\bigoplus_{p\in P}\mathbb{F}_p$ by Bergelson Host and Kra, Bergelson Taoand Ziegler and the author, respectively.

Multiple ergodic averages in abelian groups and Khintchine type recurrence