  Let $\Gamma$ be a finitely generated discrete exact group. We consider
operators on $l^2(\Gamma)$ which are composed by operators of multiplication by
a function in $l^\infty (\Gamma)$ and by the operators of left-shift by
elements of $\Gamma$. These operators generate a $C^*$-subalgebra of
$L(l^2(\Gamma))$ the elements of which we call band-dominated operators on
$\Gamma$. We study the stability of the finite sections method for
band-dominated operators with respect to a given generating system of $\Gamma$.
Our approach is based on the equivalence of the stability of a sequence and the
Fredholmness of an associated operator, and on Roe's criterion for the
Fredholmness of a band-dominated operator on a exact discrete group, which we
formulate in terms of limit operators. Special emphasis is paid to the
quasicommutator ideal of the algebra generated by the finite sections sequences
and to the stability of sequences in that algebra. For both problems, the
sequence of the discrete boundaries plays an essential role.
