  The homology groups of many natural sequences of groups
$\{G_n\}_{n=1}^{\infty}$ (e.g. general linear groups, mapping class groups,
etc.) stabilize as $n \rightarrow \infty$. Indeed, there is a well-known
machine for proving such results that goes back to early work of Quillen.
Church and Farb discovered that many sequences of groups whose homology groups
do not stabilize in the classical sense actually stabilize in some sense as
representations. They called this phenomena representation stability. We prove
that the homology groups of congruence subgroups of $GL_n(R)$ (for almost any
reasonable ring $R$) satisfy a strong version of representation stability that
we call central stability. The definition of central stability is very
different from Church-Farb's definition of representation stability (it is
defined via a universal property), but we prove that it implies representation
stability. Our main tool is a new machine for proving central stability that is
analogous to the classical homological stability machine.
