The aim of the article is to show that there are many finite extensions of
arithmetic groups which are not residually finite. Suppose G is a simple algebraic
group over the rational numbers satisfying both strong approximation, and the congruence
subgroup problem. We show that every arithmetic subgroup of G has finite
extensions which are not residually finite. More precisely, we investigate the group
H¯
2
(Z/n) =
lim
→
Γ
H
2
(Γ ,Z/n),
where Γ runs through the arithmetic subgroups of G. Elements of H¯ 2
(Z/n) correspond
to (equivalence classes of) central extensions of arithmetic groups by Z/n;
non-zero elements of H¯ 2
(Z/n) correspond to extensions which are not residually finite.
We prove that H¯ 2
(Z/n) contains infinitely many elements of order n, some of
which are invariant for the action of the arithmetic completion G[(Q) of G(Q). We
also investigate which of these (equivalence classes of) extensions lift to characteristic
zero, by determining the invariant elements in the group
H¯
2
(Zl) = lim
←t
H¯
2
(Z/l
t
).
We show that H¯ 2
(Zl)
G[(Q)
is isomorphic to Zl
c
for some positive integer c. When
G(R) has no simple components of complex type, we prove that c = b+m, where b
is the number of simple components of G(R) and m is the dimension of the centre of
a maximal compact subgroup of G(R). In all other cases, we prove upper and lower
bounds on c; our lower bound (which we believe is the correct number) is b+m
