It is well known that a finitely generated group with simply connected cones has isoperimetric functions. Papasoglu in the paper "On the Asymptotic cone of groups satisfying a quadratic isoperimetric inequality" shows that every group satisfying a quadratic isoperimetric inequality is simply connected. However as far as I understood the proof, the argument needs a finite presentation of the groups, indeed it uses a triangular presentation (page 796) that can be constructed only from a finite presentation.
Here is my question: Can the result be concluded for finitely generated groups with quadratic isoperimetric functions? And if yes why?
Thanks.
If a group is not finitely presented, it always has a non-simply-connected asymptotic cone. I won't put a full proof here, but the strategy is to pick a sequence $\{a_n\}$ of naturals corresponding to lengths of irreducible relations (here "irreducible" just means it can't be derived from shorter relations), and show the cycles in the Cayley graph corresponding to these relations converge to a non-trivial loop in any cone constructed from an ultrafilter containing $\{a_n\}$.
Thus, the statement is vacuously false for non finitely-presented groups, regardless of how you deal with the issues around isoperimetric functions on such groups.