Of the 18 infinite families of finite simple groups, I know that $\mathbb{Z}$, the infinite analog of $\mathbb{Z}_p$, isn't simple while $A_\infty$, the one for $A_n$, is. What about the 16 families of groups of Lie type? Do they have infinite analogs, and are they simple?
I know that Lie groups proper are infinite (right?). Are any of them infinite analogs to the finite groups, and if so, which ones?