Fractional isoperimetric inequalities and subgroup distortion

Isoperimetric inequalities measure the complexity of the word problem in finitely presented groups by giving a bound on the number of relators that one must apply in order to show that a word w in the given generators represents the identity. Such bounds are given in terms of the length of w, and the function describing the optimal bound is known as the Dehn function of the group. (Modulo a standard equivalence relation _, the Dehn function is an invariant of the group, not just the given finite presentation.) About seven years ago, as the result of efforts by a number of authors [9], [11], [14], it was established that for every positive integer d one can construct finitely presented groups whose Dehn function is polynomial of degree d. The question of whether or not there exist groups whose Dehn functions are of fractional degree has attracted a good deal of interest (e.g., [18], [25]). According to a theorem of M. Gromov, one does not get fractional exponents less than 2, because if a group satisfies a sub-quadratic isoperimetric inequality, then it actually satisfies a linear isoperimetric inequality (see [13], [10], [19], [22]). The main purpose of this article is to prove the following: