1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.4. Subgroups of Lie groups 193

5.4. Subgroups of Lie groups

Discrete subgroups of Lie groups provide lots of examples of amenable ac-
tions (hence exact groups). For ex.ample, let r be a discrete subgroup in
the special linear group SL(n,~) (e.g., SL(n,Z)) and let Pc SL(n,~) be
the closed subgroup of upper triangular matrices with positive diagonals. It
turns out the left multiplication action of r on the compact homogeneous
space SL(n,~)/P is amenable. (Note that SL(n,~)/P is compact since it
is homeomorphic to SO ( n, ~).)
More generally, let G be a real semisimple Lie group, or any other (sec-
ond countable) locally compact group which admits an Iwasawa decompo-
sition G = KAN into closed subgroups with K compact, A abelian and
N nilpotent, such that A normalizes N. (When G = SL(n,~), one can
take K = SO(n,~), A to be the diagonal matrices with positive entries and
determinant 1, and N to be the upper triangular matrices with 1 's on the
diagonal.) The closed subgroup P = AN is solvable and hence is amenable
(as a locally compact group^13 ). Then, as we will prove below, any discrete
subgroup r ::::; G (i.e., r is discrete in the relative topology) acts amenably
on the compact homogeneous space X = G / P; this result is essentially due
to Connes.

Theorem 5.4.1. Let G be a second countable locally compact group, r ::::; G

be a discrete subgroup and P C G be a closed amenable subgroup such that

X = G / P is compact. Then, the left multiplication action of r on X is

amenable.

Proof. It is well known ([10, Theorem 3.4.1] applied to a compact subset

of G which surjects onto X) that there exists a regular Borel cross section

rT: X ---+ G; that is, O" is a Borel map such that <Y(x)P = x for all x E X and

O"(X) is pre-compact. Likewise, there is a Borel fundamental domain Y CG

for the f action -i.e., Y is BoreI and G decomposes into the disjoint union

G = Us Er s Y. It follows that

Prob(G) 3 fl f-+ (fi(sY))sEr E Prob(f)

is a r-equivariant continuous map. Now, let a finite subset E c rands> 0

be given. Then,

E = {<Y(sx)-^1 so-(x): s EE, x EX}

is a pre-compact subset in P. Since Pis amenable, there exists v E Prob(P)

such that llg.v - vii < s for all g EE .. We regard v as a measure on G and

13For information on amenability of locally compact groups, see [139].