In Sapir, Mark V.; Birget, Jean-Camille; Rips, Eliyahu Isoperimetric and isodiametric functions of groups. Ann. of Math. (2) 156 (2002), no. 2, 345–466 we proved that P=NP if and only if the word problem in every group with polynomial Dehn function can be solved in polynomial time by a deterministic Turing machine. Thus to show that P=NP one "only" needs to find an algebraic description of finitely presented groups with polynomial Dehn functions (similar to Gromov's description of groups with polynomial growth functions).
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