Is it obligatory for the Imam to remain silent after Al-Fatihah so that the worshipper can recite Al-Fatihah?

The Imam is not obligated to be silent after Al-Fatihah, nor is it recommended for him to do so according to the majority of scholars. The pause mentioned in some hadiths after Al-Fatihah is a brief pause for separation and not for the congregant to recite Al-Fatihah. Scholars have differed regarding the number of pauses for the Imam: some said there should be no pause at all, some said one pause for the opening supplication (istiftah), and some said two pauses: one for the opening supplication and another before bowing (rukūʿ).