A theorem of Bass-Milnor-Serre says that for n > 2 every finite dimensional representation of SL_n(Z) virtually extends to a representation of SL_n(R) -- meaning there is a representation of SL_n(R) that agrees with it along a finite index subgroup of SL_n(Z). Moreover this theorem is fairly tight in the sense that if we remove any of the phrases "n > 2", "finite dimensional", or "virtually" then this theorem fails spectacularly. SL_\infty(Z) has no non-trivial finite dimensional representations and no finite index subgroups, but nevertheless I will formulate an infinite-rank version of this theorem. Speaker(s): Nate Harman (University of Michigan)