Jordan's theorem on finite subgroups of $GL_n(C)$ asserts that each such group is contained in an algebraic subgroup of $GL_n$ which is toral-by-finite and of bounded complexity. As such, it exemplifies the idea of trying to relate various kinds of linear groups to linear algebraic "envelopes" of some kind. I will discuss some variations on this theme.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3472
[3] http://www.mpim-bonn.mpg.de/node/4751