Stable splittings, spaces of representations and almost commuting elements in Lie groups

In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits $\text{Rep}({\mathbb{Z}}^n,SU)$, $\text{Rep}({\mathbb{Z}}^n,U)$ and $\text{Rep}({\mathbb{Z}}^n,Sp)$ are explicitly described as finite products of Eilenberg-MacLane spaces.