Abstract

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 F n 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.

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