Can they hospitalize people for cutting and si in america?

That depends. Many: People w suicidal thoughts do not require hospitalization. However if the person is afraid of acting on suicidal thoughts, they want to kill themself & have thoughts of how to do it - in the us, more often than not, it would be reason for admission. As to cutting - if it was a suicide attempt or a cry for help these could be reason to admit. Chronic cutters with no si often don't need admission.