We choose a normal subgroup [GAMMA]' [??] [GAMMA] of

finite index that has no elliptic point nor irregular cusp.

where [??] and [??] are taken for every

finite index set K such that [lambda] = [[??].sub.i[member of]K][[lambda].sub.i], respectively.

Comparing homomorphisms into H I An with homomorphisms into H [??] [S.sub.n] gives information on the embedding of

finite index subgroups in large groups.

There exists a subgroup H of

finite index in G, which does not intersect F, and such that the right cosets Hu, u [member of] F, are pairwise disjoint.

Now the general case for

finite index to prove that K is a field follows as in [6].

Since A is finitely generated torsion-free abelian, C is a direct factor of a subgroup [A.sub.1] of A of

finite index; that is, [A.sub.1] = C X H, where [A : [A.sub.1]] = n < [infinity].

Moreover, the tower !G.sub.k^ = !(!summation of^X).sub.k^, !S.sup.!n.sub.!Alpha^^^ has the property that for each k, the image of !G.sub.k+1^ has

finite index in !G.sub.k^.

For any field L with F [subset] L [subset] [??], the group G(L) is a discrete subgroup of G([A.sub.L]) if and only if A(K) [subset] A(L) is of

finite index.

Secondly, if C is a subfactor of

finite index in M, then its center is atomic, but its relative commutant is finite dimensional.

In this paper we deal only with regular languages, for which [[less than or equal to].sub.L] has

finite index. The reason is that for a regular language L there are only finitely many languages of the form [x.sup.-1] [Ly.sup.-1] = {u [member of] [A.sup.*] | xuy [member of] L}.

Then [G'.sub.0] is a

finite index solvable normal subgroup of G'.

If G is infinite amenable group and H contains a

finite index normal subgroup N with N [intersection] A = {1}, then (G, H, A) [member of] A'.