There isn't a specific prayer book of Islam,.. prayers are usually memorized. Islamic prayers are often written down in general prayer books for the purpose of assisting Muslims to learn them.
The person who leads prayers in a synagogue is called a rabbi.
Sheliah Tzibbur - messenger of the community. The Rabbi does not need to be the one leading the prayers.
Synagogue comes from a Greek word meaning "assembly," and may have come about in ancient times and been influenced by the Hebrew term בית כנסת, meaning "house of assembly." Other Hebrew names...
You may be thinking of the book currently called The Liturgy of the Hours. It contains the prayers called matins, lauds, vespers, compline, etc. It is commonly called the Breviary..
Indeed,...