What does absolutely mean?

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It can mean all sorts of things. Often it is used as perhaps the most extreme modifier
A: Why did you get married to her?
B: I was absolutely besotted (in love with her head over heels). 完全に 極度に

In the latter sense it can also be used to express mean extreme disagreement
A: People should get married young.
B: I absolutely disagree. ぜんぜん賛成できません。

But on its own, it is often used as a means of agreement.
A: We need to take more care of the environment.
B: (I agree) Absolutely! まったく

In the latter case it can be used a little sarcastically, by English people, to invoke agreement that is so extreme that is should not be worth saying.
A: We should all try to do our best at this. (皆、頑張りましょうね）
B: Absolutely...（それはそうだね）