Subset currents on free groups. (Q376290)

Let \(F_N\) be the free group on \(N\) generators. The geodesic currents on \(F_N\), denoted \(\text{Curr}(F_N)\), is the set of all positive, \(F_N\)-invariant, locally finite Borel measures defined on the set of two-element subsets of \(\partial F_N\). This set was used in the study of properties of the outer space of \(F_N\). The elements of \(\text{Curr}(F_N)\) are the analogues of the conjugacy classes of elements in \(F_N\). In the paper under review, the authors introduce a generalization of \(\text{Curr}(F_N)\), the subset currents, denoted by \(\mathcal S\text{Curr}(F_N)\), to be the set of all positive, \(F_N\)-invariant, locally finite Borel measures on the set \(\mathfrak C_N\) of all subset of \(\partial F_N\) containing at least two elements. The authors generalize the constructions on \(\text{Curr}(F_N)\) to \(\mathcal S\text{Curr}(F_N)\) and show that \(\mathcal S\text{Curr}(F_N)\) has much richer properties. As a first remark, the elements of \(\mathcal S\text{Curr}(F_N)\) are the analogues of conjugacy classes of finitely generated subgroups of \(F_N\). The space \(\mathcal S\text{Curr}(F_N)\) admits the weak \(*\)-topology of convergence of integrals of continuous functions with compact support. The space \(\mathcal S\text{Curr}(F_N)\) admits an action of \(\text{Out}(F_N)\). For \(\mu\in\mathcal S\text{Curr}(F_N)\), \(U\in\mathfrak C_N\), and \(\varphi\in\text{Out}(F_N)\), \(\varphi\mu(U)=\mu(\varphi^{-1}(U))\). That makes \(\text{Curr}(F_N)\) a closed \(\text{Out}(F_N)\)-invariant subset. Let \(H\) be a finitely generated nontrivial subgroup of \(F_N\), the authors define the counting measure of \(H\). Let \(\text{Comm}_{F_N}(H)\) be the virtual normalizer of \(H\). If \(H=\text{Comm}_{F_N}(H)\), define \(\eta_H\in\mathcal S\text{Curr}(F_N)\) to be the sum, over the conjugates of \(H\), of the \(\delta\) measures of the limit sets of the conjugates. In general, if \(H_0\) is the virtual normalizer of \(H\) and \(H\) has index \(m\) in \(H_0\), then \(\eta_H=m\eta_{H_0}\). Positive multiples of the measures \(\eta_H\) are called rational subset currents. The set of rational subset currents is a dense subset of \(\mathcal S\text{Curr}(F_N)\), generalizing the corresponding result for geodesic currents. Using \(\eta_H\), the authors define a pairing on \(\text{cv}_N\times\mathcal S\text{Curr}(F_N)\) with values in \(\mathbb R_{\geq 0}\), where \(\text{cv}_N\) is the outer space of \(F_N\). If \(T\) is a tree, representing an element of \(\text{cv}_N\) and \(\eta_H\) a counting measure, then \(\langle T,\eta_H\rangle=\text{vol}(T_H/H)\), where \(T_H\) is a minimal \(H\)-invariant subtree of \(T\). This covolume pairing generalizes the corresponding pairing on \(\text{Curr}(F_N)\). The only difference is that the new pairing does not extend to the compactification of \(\text{cv}_N\), as it does the old one. For a finitely generated subgroup \(F\) of \(F_N\), define the reduced rank \(\overline{\text{rk}}(F)\) to be \(\text{rk}F-1\), if \(F\) is not trivial and \(0\) if it is trivial. Then the function \(\overline{\text{rk}}\) defined on \(\mathcal S\text{Curr}(F_N)\) by the rule \(\overline{\text{rk}}(\eta_H)=\overline{\text{rk}}(H)\) is an \(\mathbb R_{\geq 0}\)-linear, \(\text{Out}(F_N)\)-invariant functional. Finally, the authors ask a series of questions that lead to further investigating the properties of the subset currents, as for example the extension of the constructions to general word hyperbolic groups.