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Subset currents on free groups. (English) Zbl 1288.20053
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.

MSC:
 20F65 Geometric group theory 20E05 Free nonabelian groups 20F28 Automorphism groups of groups 20E08 Groups acting on trees 57M07 Topological methods in group theory 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37B10 Symbolic dynamics 37E25 Dynamical systems involving maps of trees and graphs
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