346 12. Approximation Properties for GroupsTherefore,§2^1 (B1 U B2) c Bo U Bi,
§1(B3 U B4) c Bo U B3,
§2(B3 U B4) c Bo U B4.μ(B1) + 1-i(B2) = μ(B1 U B2) S μ(Bo U B2) + 28 S μ(B2) + 82 + 28
and hence μ(B1) s c5^2 +2c5. The same inequality holds for the other μ(Bk)'s.
It follows that
4
1 = μ(1'^2 \ {o}) = Lμ(Bk) S 582 + 86 < 108 < 1,
k=O
which is a contradiction. D
Remark 12.1.11. Actually, one can show that whenever r c 81(2, Z) is
nonamenable, the inclusion (Z^2 c Z^2 ><1 I') has relative property (T). Indeed,
this follows from the following fact (combined with the proof above): There
is no sequence (μn) of regular Borel probability measures on 1'^2 with the
following properties:
(1) μn({o}) = 0 for every n;
(2) the sequence (μn) converges weakly to the Dirac measure 60 at o;
(3) the sequence (μn) is approximately I'-invariant, i.e., for all g EI'
limsup{lμn(§B) - μn(B)I: BC 1'^2 a Borel subset}= 0.
n
To show that no such measures exist, we regard the μn's as probability
measures on ']'^2 \ { o }. After blowing up the hole and patching it with
JRP^1 ~ 'TI', the resulting space M = ('TI'^2 \ { o}) U JRP^1 is a compact manifold.
Moreover, the action of SL(2, Z) on ']'^2 \ { o} naturally extends to M. Now,
μn is a probability measure on a compact space M and hence the sequence
μn has a limit point μ. By assumption, μ is supported on JRP^1 and is r-
invariant. However, this is impossible since r is nonamenable and JRP^1 is
r-amenable (see Theorem 5.4.1 and Example E.10).
Now we turn to the bounded generation trick. We define subgroups
G, H1, H2 c SL(3, Z) by
G = [: : ~i ' H1 = [~ ~ :] and H2 = [~ ~ ~l ·
001 001 **1
Note that G normalizes both H1 and H2.
Lemma 12.1.12. SL(3, Z) = H1H2H1H2G.