  This is the third paper in our series of five in which we test the Master
Constraint Programme for solving the Hamiltonian constraint in Loop Quantum
Gravity. In this work we analyze models which, despite the fact that the phase
space is finite dimensional, are much more complicated than in the second
paper: These are systems with an $SL(2,\Rl)$ gauge symmetry and the
complications arise because non -- compact semisimple Lie groups are not
amenable (have no finite translation invariant measure). This leads to severe
obstacles in the refined algebraic quantization programme (group averaging) and
we see a trace of that in the fact that the spectrum of the Master Constraint
does not contain the point zero. However, the minimum of the spectrum is of
order $\hbar^2$ which can be interpreted as a normal ordering constant arising
from first class constraints (while second class systems lead to $\hbar$ normal
ordering constants). The physical Hilbert space can then be be obtained after
subtracting this normal ordering correction.
