Genericity of Filling Elements

An element of a finitely generated non-Abelian free group F(X) is said to be filling if that element has positive translation length in every very small action of F(X) on an $\mathbb{R}$-tree. We give a proof that the set of filling elements of F(X) is exponentially F(X)-generic in the sense of Arzhantseva and Ol'shanskii. We also provide an algebraic sufficient condition for an element to be filling and show that there exists an exponentially F(X)-generic subset of filling elements whose membership problem is solvable in linear time.