Why did the gap between rich ans poor in Latin America grow after independence.

In a way, it's always easier to increase this gap than to diminish it, so if no measures are taken, this will be the result. In the case of Latin America two factors specifically came into play, but not the only: the rich became even richer, as they purchased and owned more and more land, and the poor needed credits to survive, so they ended up with huge debts.