Poll: They're ____ criminals.
They're ____ criminals.
Ok, you can't say "famous criminals" because that is using the word 'famous' wrong and is just silly. You also can't say "infamous criminals" because that is redundant. "notorious" is the only choice here.
Of course criminals can be famous, too. A "criminal" is condemned so by the law. If the people, however, do not agree with the law and the criminal as a kind of hero, say a vigilante like DareDevil, than that "criminal" is not infamous for any bad actions. The people wouldn't call them a criminal, but a news paper would have to write "the famous criminal", because not matter how heroic, a criminal they are.
"Infamous criminal" is by no means redundant. A criminal is not famous or infamous, if they aren't even known by anybody. So they become infamous, when they get widespread attention for their actions.
And "notorious" simply means "well known" and is in principle a synonym to both famous and infamous. However, in recent decades it has gained a very negative context, much stronger than infamous. A bad celebrity, or their action can be infamous. But notorious has a stronger ring to it and creates an aura of fear and violence. However, that is just how the originally neutral word has been painted only be the way it is regularily used.