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Volume equivalence of subgroups of free groups

Volume equivalence of subgroups of free groups,10.1016/j.jalgebra.2010.03.027,Journal of Algebra,Donghi Lee,Enric Ventura

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Volume equivalence of subgroups of free groups  
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Let Fn 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 Fn are called volume equivalent, if for every free and discrete isometric action of Fn on an R-tree T, we have vol(TH/H)=vol(TK/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.
Journal: Journal of Algebra - J ALGEBRA , vol. 324, no. 2, pp. 195-217, 2010
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