Currents on free groups.

*(English)*Zbl 1110.20034
Grigorchuk, Rostislav (ed.) et al., Topological and asymptotic aspects of group theory. AMS special session on probabilistic and asymptotic aspects of group theory, Athens, OH, USA, March 26–27, 2004 and the AMS special session on topological aspects of group theory, Nashville, TN, USA, October 16–17, 2004. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3756-7/pbk). Contemporary Mathematics 394, 149-176 (2006).

Let \(G\) be a word hyperbolic group, \(\partial G\) its (hyperbolic) boundary and \(\partial^2G=\{(\zeta,\xi)\), \(\zeta,\xi\in\partial G\), \(\zeta\neq\xi\}\). A ‘geodesic current’ on \(G\) is a positive locally finite Borel measure on \(\partial^2G\) which is \(G\)-invariant. Let \(\text{Curr}(G)\) be the space of all geodesic currents on \(G\). There are equivalent definitions of the space \(\text{Curr}(G)\) [see F. Bonahon, in Arboreal group theory, Berkeley/CA 1988, Publ., Math. Sci. Res. Inst. 19, 143-168 (1991; Zbl 0772.57004)].

The aim of this paper is, mainly, to do a systematic exposition of previous results on geodesic currents on a free non-Abelian group. Some of these results have appeared previously [in loc. cit.; I. Kapovich, Int. J. Algebra Comput. 15, No. 5-6, 939-969 (2005; Zbl 1110.20031 above); R. Martin, Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups, PhD Thesis (1995)].

Here it are studied induced actions of \(\text{Out}(F)\) and \(\operatorname{Aut}(F)\) on \(\text{Curr}(F)\). In Proposition 4.5 it is shown that \(\text{Curr}(F)\) is homeomorphic to some other spaces. This marks out a relation of \(\text{Curr}(F)\) and the Culler-Vogtmann Outer space.

Let \(\text{FLen}(F)\) be the space of all hyperbolic length functions \(\ell\colon F\to\mathbb{R}\) corresponding to free and discrete actions of \(F\) on \(\mathbb{R}\)-trees, then it is defined a map \(I\colon\text{FLen}(F)\times\text{Curr}(F)\to\mathbb{R}\), this “intersection form” has good properties (Proposition 5.9 and Theorem 6.2). Especially the restriction of this map, with respect to the second argument, to the rational currents (Definition 5.1) is the length function \(\ell\). Also it is proved that the value of the intersection map on the uniform currents on \(F\) corresponding to a free basis \(A\) of \(F\) (Definition 7.1) is equal to the generic stretching factor (Definition 7.5) of \(\ell\) with respect to the free basis \(A\) (Proposition 9.1).

The paper concludes with a generalization of the action of \(\operatorname{Aut}(F)\) on \(\text{Curr}(F)\) in the following sense: Let \(\varphi\colon F_1\to F_2\) be a monomorphism between the finitely generated non-Abelian free groups \(F_1\) and \(F_2\), then there exists a unique continuous linear map \(\varphi_*\colon\text{Curr}(F_1)\to\text{Curr}(F_2)\) “induced” by \(\varphi\). This puts in a correct base a previous claim of R. Martin in his above mentioned thesis. Finally it is introduced the concept of the Translation Equivalence of Currents (two currents \(\nu_1,\nu_2\in\text{Curr}(F)\) are said to be translation equivalent if \(I(\ell,\nu_1)=I(\ell,\nu_2)\) for every \(\ell\in\text{FLen}(F)\)) and it is given a partial case of translation equivalent currents.

For the entire collection see [Zbl 1085.20501].

The aim of this paper is, mainly, to do a systematic exposition of previous results on geodesic currents on a free non-Abelian group. Some of these results have appeared previously [in loc. cit.; I. Kapovich, Int. J. Algebra Comput. 15, No. 5-6, 939-969 (2005; Zbl 1110.20031 above); R. Martin, Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups, PhD Thesis (1995)].

Here it are studied induced actions of \(\text{Out}(F)\) and \(\operatorname{Aut}(F)\) on \(\text{Curr}(F)\). In Proposition 4.5 it is shown that \(\text{Curr}(F)\) is homeomorphic to some other spaces. This marks out a relation of \(\text{Curr}(F)\) and the Culler-Vogtmann Outer space.

Let \(\text{FLen}(F)\) be the space of all hyperbolic length functions \(\ell\colon F\to\mathbb{R}\) corresponding to free and discrete actions of \(F\) on \(\mathbb{R}\)-trees, then it is defined a map \(I\colon\text{FLen}(F)\times\text{Curr}(F)\to\mathbb{R}\), this “intersection form” has good properties (Proposition 5.9 and Theorem 6.2). Especially the restriction of this map, with respect to the second argument, to the rational currents (Definition 5.1) is the length function \(\ell\). Also it is proved that the value of the intersection map on the uniform currents on \(F\) corresponding to a free basis \(A\) of \(F\) (Definition 7.1) is equal to the generic stretching factor (Definition 7.5) of \(\ell\) with respect to the free basis \(A\) (Proposition 9.1).

The paper concludes with a generalization of the action of \(\operatorname{Aut}(F)\) on \(\text{Curr}(F)\) in the following sense: Let \(\varphi\colon F_1\to F_2\) be a monomorphism between the finitely generated non-Abelian free groups \(F_1\) and \(F_2\), then there exists a unique continuous linear map \(\varphi_*\colon\text{Curr}(F_1)\to\text{Curr}(F_2)\) “induced” by \(\varphi\). This puts in a correct base a previous claim of R. Martin in his above mentioned thesis. Finally it is introduced the concept of the Translation Equivalence of Currents (two currents \(\nu_1,\nu_2\in\text{Curr}(F)\) are said to be translation equivalent if \(I(\ell,\nu_1)=I(\ell,\nu_2)\) for every \(\ell\in\text{FLen}(F)\)) and it is given a partial case of translation equivalent currents.

For the entire collection see [Zbl 1085.20501].

Reviewer: Dimitrios Varsos (Athenai)

##### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

20F65 | Geometric group theory |

20E05 | Free nonabelian groups |

20F28 | Automorphism groups of groups |

20E36 | Automorphisms of infinite groups |

20F05 | Generators, relations, and presentations of groups |

57M05 | Fundamental group, presentations, free differential calculus |