Is it true that the Baumslag-Solitar groups, say, $BS(1,n)$, $|n|\ge 2$, are finitely presented groups with largest Dehn functions (namely, exponential growth) known to be inside finitely presented groups with quadratic Dehn functions? 



What is the largest known Dehn function of f.p. subgroup of a f.p. group with quadratic Dehn function?
Yes, the highest known Dehn function of a subgroup of a finitely presented group with quadratic Dehn function is exponential. The example was found by Yves Cornulier and Romain Tessera in 
Metabelian groups with quadratic Dehn function and Baumslag-Solitar groups. 
Confluentes Math. 2 (2010), no. 4, 431–443.


