They are sometimes called stable groups, though this term normally means something quite different in model theory.
This is a priori surprising, given that they are generally infinite-dimensional, constructed as limits of groups with finite-dimensional representations.
Each family of classical groups forms a direct system, via inclusion of matrices in the upper left corner, such as
Bott periodicity computes the homotopy of the stable unitary group and stable orthogonal group.
The Whitehead group of a ring (the first K-group) can be defined in terms of