Volume equivalence of subgroups of free groups

Let F(n) be a free group of finite rank n >= 2. Due to Kapovich, Levitt, Schupp and Shpilrain (2007) [1], two finitely generated subgroups H and K of F(n) are called volume equivalent, if for every free and discrete isometric action of Fn on an R-tree T, we have vol(T(H)/H) = vol(T(K)/K). We give a more algebraic and combinatorial characterization of volume equivalence and discuss a counterexample of volume equivalence in order to justify our characterization. We also provide a specific example to answer a question of Kapovich, Levitt, Schupp and Shpilrain in the negative: volume equivalence does not imply equality in rank. (C) 2010 Elsevier Inc. All rights reserved.