residually finite group
Imagine an infinite group, possibly huge and complicated. You want to detect a particular nonidentity element — to be sure it really is not the identity. Residual finiteness says you can always do so by shrinking the group down to a finite one in which that element still survives. No nontrivial element is invisible to all finite quotients; every distinction in the group is witnessed by some finite picture.
Precisely, a group G is residually finite if for every g ≠ 1 there is a finite group F and a homomorphism φ : G to F with φ(g) ≠ 1. Equivalently, the intersection of all finite-index normal subgroups is trivial, or equivalently G embeds into the inverse limit of its finite quotients (its profinite completion) with trivial kernel. The property passes to subgroups and to finite direct products, and is inherited by any group embedding into a residually finite group.
Residual finiteness is both common and powerful. Finitely generated linear groups are residually finite (Malcev's theorem), so finitely generated free groups, polycyclic groups, surface groups, and many others qualify. Crucially, Mal'cev observed that a finitely presented residually finite group has solvable word problem — one enumerates relations to confirm w = 1, and enumerates finite quotients to confirm w ≠ 1, and one search must halt. Not all groups are residually finite, though: simple infinite groups and BS(2, 3) are not.
Z is residually finite: any nonzero integer n maps to a nonzero class in Z/mZ for any m not dividing n. More generally GL(k, Z) is residually finite via reduction modulo primes, the maps GL(k, Z) to GL(k, Z/pZ).
Z and GL(k, Z) are residually finite by reduction modulo m or modulo p.
Residual finiteness is not a quasi-isometry invariant: there exist quasi-isometric pairs of finitely presented groups, one residually finite and the other not. It is, however, a commensurability invariant.