Multiple ergodic averages in abelian groups and Khintchine type recurrence

Let $G$ be a countable abelian group. We study ergodic averages associated with 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 characteristic factor for these averages is a factor of a $2$-step nilpotent homogeneous space. As an application we derive a Khintchine type recurrence result. In particular, 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 finite index in $G$, then for every $E\subset G$ and $\varepsilon>0$ the set $$\{g\in G : d(E\cap E-ag\cap E-bg\cap E-(a+b)g)\geq d(E)^4-\varepsilon\}$$ 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 Tao and Ziegler and the author, respectively.