Volume equivalence of subgroups of free groups. (Q986063)

Let \(F_n\) be the free group of finite rank \(n\geq 2\). \textit{I. Kapovich, G. Levitt, P. Schupp} and \textit{V. Shpilrain}, [Trans. Am. Math. Soc. 359, No. 4, 1527-1546 (2007; Zbl 1119.20037)], introduced the notion of the translation equivalence. The elements \(g,h\in F_n\) are called translation equivalent if for every free and discrete isometric action of \(F_n\) on an \(\mathbb{R}\)-tree \(T\) we have \(\inf_{x\in T}d(x,gx)=\inf_{x\in T}d(x,hx)\). In the above mentioned paper several characterizations of translation equivalence are given. There also two natural generalizations of the notion of translation equivalence are introduced. Two elements \(g,h\in F_n\) are called boundedly translation equivalent if there is \(C>0\) such that \(\frac{1}{C}\leq\frac{\|\varphi(g)\|}{\|\varphi(h)\|}\leq C\) for every automorphism \(\varphi\) of \(F_n\). 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 \(\mathbb{R}\)-tree \(T\) we have \(\text{vol}(T_H/H)=\text{vol}(T_K/K)\). -- Here \(T_H\) is the unique minimal \(H\)-invariant subtree of \(T\); thus \(T_H/H\) is a finite graph whose edges inherit the same lengths as their lifts to \(T\). The volume \(\text{vol}(T_H/H)\) is the sum of the lengths of the edges of this graph. Let \(F_n\) be the free group of finite rank \(n\geq 2\) with basis \(\Sigma\). Every finitely generated subgroup \(H\) of \(F_n\) has a unique finite graphical representation \(\Gamma_H\) up to graph isomorphism. The graph \(\Gamma_H\) is a cyclically reduced \(\Sigma\)-labeled directed graph which depends only on the conjugacy class \([H]\). For \(S\subset\Sigma^\pm\), the capacity \(\text{cap}(\Gamma_H;S)\) of \(S\) is the number of vertices \(v\) of \(\Gamma_H\) such that the set of labels of incoming edges of \(v\) meets both \(S\) and its complement \(S^c\). The main results of this paper are: Theorem A. (i) For \(n\geq 3\), two finitely generated subgroups \(H\) and \(K\) of \(F_n\) are volume equivalent if and only if \(E(\Gamma_{\varphi(H)})=E(\Gamma_{\varphi(K)})\) for every automorphism \(\varphi\) of \(F_n\). (Here \(E(\Gamma_{\varphi(H)})\) means the number of the edges of the graph \(\Gamma_{\varphi(H)}\).) (ii) Two finitely generated subgroups \(H\) and \(K\) of \(F_2=F(a,b)\) are volume equivalent if and only if \(E(\Gamma_{\varphi(H)})=E(\Gamma_{\varphi(K)})\) and \(\text{cap}(\Gamma_{\varphi(H)};\{a^{\pm 1}\})=\text{cap}(\Gamma_{\varphi(K)};\{a^{\pm 1}\})\) for every automorphism \(\varphi\) of \(F_2\). The proof of this Theorem is a modification and generalization of the proof of Theorem A in the above mentioned paper. As it is stated in the Theorem A above in the case of the free rank 2 an extra condition on capacities is needed. This condition cannot be removed. Theorem B. Let \(H=\langle[a,b]\rangle\) and \(K=\langle a,b^2,bab^{-1}\rangle\) be subgroups of \(F(a,b)\). Then we have \(E(\Gamma_{\varphi(H)})=E(\Gamma_{\varphi(K)})\) for every automorphism \(\varphi\) of \(F(a,b)\), but \(H\) and \(K\) are not volume equivalent. The main part of the paper is devoted to prove, giving an example, (Theorem C in the paper) that the volume equivalence of two subgroups of \(F_n\) does not imply that they have the same rank.