Like many of the people here, I have a different view of real life than most of the population of the planet. Lattices like $SL(n, Z)$ and $Sp(2g, Z)$ come up frequently in topology (often via the study of the mapping class group, see, eg, Farb-Margalit's book, available online), and in algebraic geometry (including Deligne's proof of the Weil conjecture, see the considerable oevre of Nick Katz). Also in the study of singularities of differential equations (all three of the above are related). This has much to do with number theory (eg, recent work on Sato-Tate, though I will not embarrass myself further by talking about things I have no idea of). Physicists (in the weak sense, that is, string theorists, who are primarily mathematicians in my view) have used much of the above in the study of Calabi-Yau manifolds.

I believe that the applications to real real life (such as building bridges and constructing iPhones), you might be, sadly, out of luck.