Being of type FP2 is an algebraic shadow of being finitely presented. A long standing question was whether these two classes are equivalent. This was shown to be false in the work of Bestvina and Brady. More recently, there are many new examples of groups of type FP2 coming with various interesting properties. I will begin with an introduction to the finiteness property FP2. I will end by giving a construction to find groups that are of type FP2(F) for all fields F but not FP2(ℤ).
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