TITLE: Center of a finitely generated group
QUESTION [14 upvotes]: Could you give me an example (with proof) of a finitely generated group with a not finitely generated center?

REPLY [2 votes]: V. N. Remeslennikov found a finitely presented group whose center is not finitely generated.  But I didn't check his construction.
Remeslennikov, V. N.
A finitely presented group whose center is not finitely generated. (Russian) 
Algebra i Logika 13 (1974), no. 4, 450–459, 488. 
English translation: Algebra and Logic 13 (1974), no. 4, 258–264 (1975)