I'm not going to read all of the posts and just get to the original question. Muhammad, was a warrior chieftain, a thief, diddled with a child and married her when she turned nine years of age. He believed that his religious ideology (a mix of the Old Testament and his own ideas), was what those who followed him, must believe. Any who refused to get onboard with his warped religious ideology were either enslaved or killed. His stated goal was to conquer the world. One of the tenets of his religious ideology is the da'wah, the migration of Muslims to "infidel" (that's you, I and others) lands for the purpose of changing those infidel lands into Islamic lands, by conversion and by violence. Once the infidel lands would be conquered, the non-believers had the following choices: Convert, be enslaved, be killed or flee.
For many centuries Muslims have repeatedly tried to conquer infidel lands by "the sword." Eventually, they realized that they couldn't succeed against the infidels via outright large scale wars. Fast forward to the twentieth century. A worldwide tactic of guerilla warfare ensued in infidel lands. Small and medium groups of fighters, claiming they were fighting for freedom, but were carrying out their atrocities for the sole purpose of gaining territory. Entering into the twenty-first century, the Islamists got a break to move forward on their goal of conquering infidel lands, they were actually "invited" by the naive and clueless infidels to emigrate there. They realized that they didn't have to send large armies against the infidels and shed huge numbers of their followers. They just had to accept the invitation and flood combat age males into those nations and tear them apart from within. So, as to why they stay in the infidel lands and not go back, their goal isn't to return but create more Islamic nations. They'll never go back willingly.
