And a very large population of Christians and Jews who built and maintained these public works.
Mind you, the Moors that ruled southern Spain were in many ways quite distinct from the Arabs that conquered from the east. It's very likely the Moors back then spoke their own language as much as Arabic.
In my opinion, the reason why Islam grew so fast just after Mohammaed's time was just as much economic as political--after all, the Arabian peninsula was a crossroads of trade between Europe and Asia and down the east coast of Africa. Because the Arabs effectively cornered that east-west trade small wonder why the Europeans looked fervently for a way to NOT having to deal with the Arabs when trading for goods from Asia, hence the numerous exploration voyages from the 1400's on by European powers.
Ironically, many of the greatest achievements of Islam were NOT done the Arabs. Saladin was a Kurd that beat back the Crusaders from what is now Israel and Lebanon; it was the Ottoman Turks that finally overran Constantinople and put an end to the Byzantine Empire; and it was Moors of northwest Africa that created the great Islamic cities of southern Spain. I think within 20 years Bahrain, Qatar and the United Arab Emirates will be the richest countries in the Persian Gulf, mostly because all three countries have figured out that their geographical location could make them the warehouse of Eurasia quite easily and help revive the ancient trading routes through the Arabian Peninsula.