I'm having a lot of trouble nailing down your question.  After reading it several times, I think you mean that you have an opportunity to go on to study a PhD in geometric group theory at your current university.  (Is it specifically relevant that you are going abroad tomorrow?  How does this fit in with the rest?)  From this I am guessing that you are not in the US, because in the US one would not have a position in "geometric group theory" specifically; one would just apply to the entire math department.  It could be helpful to know what country you are studying in.

So I think you are really asking whether you should go on to do a PhD?  You say that your motivation is to do mathematical research: for that you essentially need a PhD, yes.  


  Along the way I created the vision that I would finish my master degree as a dedicated student in algebra.


Yes, okay, you're studying algebra.  I feel like I'm missing some nuance here: what's the "vision"?


  As mentioned my dream is to do my own research, but more generally/important: I want to develop new ideas and be dedicated to something and I would love to follow/explore my passion.


Again, I feel like I'm missing a nuance.  Doing your own research means precisely to developing your new ideas and being dedicated to something.  Doesn't everyone want to "follow/explore their passion"?  That comes off sounding mean, which is not my intent: rather, I feel like you're trying to say something here and I'm missing it.


  I don't know much about geometric group theory, but I do know that I love group theory. So I would really like to know if this is an opportunity or if I should just forget it.


It seems clear that a PhD in geometric group theory is an indeed an opportunity to do everything that you said you wanted to do...in geometric group theory, which is a subfield of the field that you say you like.  Whether you will like GGT specifically seems best answered by learning some and finding out.  If the PhD program is so specifically invested in GGT that if you find out you don't want to do GGT per se then you'll need to leave the program, then you should study GGT on your own / during your master's degree first in order to find out whether it's interesting enough to you to spend years of your life on.  (I think GGT is really interesting.  If I hadn't done arithmetic geometry, I think I would have had fun with it.  So I assure you that there is nothing wrong with that subsubfield of mathematics!)


  I don't want to pursue a PhD for anything in this world. I just want to stay focus on creating a meaningful work life around my passion.


Huh?  This time I don't understand at all.  Please elaborate/rephrase.

Career advice – mathematics
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I use zend 

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