  We introduce an $SU(1,1)$ algebraic approach to study the $(2+1)$-Dirac
oscillator in the presence of the Aharonov-Casher effect coupled to an external
electromagnetic field in the Minkowski spacetime and the cosmic string
spacetime. This approach is based on a quantum mechanics factorization method
that allows us to obtain the $su(1,1)$ algebra generators, the energy spectrum
and the eigenfunctions. We obtain the coherent states and their temporal
evolution for each spinor component of this problem. Finally, for these
problems, we calculate some matrix elements and the Schr\"odinger uncertainty
relationship for two general $SU(1,1)$ operators.
