Groups with fractionally exponential subgroup growth

We prove that, if the subgroup growth of a finitely generated metabelian group G is not polynomial, then it is at least c^{n 1 d} for some positive integer d (where c>1 is a suitable constant). For each integer d>1 we construct a finitely presented metabelian group whose subgroup growth is approximately c^{n 1 d}. Finally, we establish a sharp upper bound for the subnormal subgroup growth of finitely presented soluble groups, and derive a new necessary condition for a metabelian group to be finitely presented. Our methods involve results from algebraic geometry and the geometry of numbers, as well as Golod-Safarevic type inequalities.