Non-unique ergodicity, observers’ topology and the dual algebraic lamination for \(\mathbb{R}\)-trees.

*(English)*Zbl 1197.20020Summary: Let \(T\) be an \(\mathbb{R}\)-tree with a very small action of a free group \(F_N\) which has dense orbits. Such a tree \(T\) or its metric completion \(\overline T\) are not locally compact. However, if one adds the Gromov boundary \(\partial T\) to \(\overline T\), then there is a coarser ‘observers’ topology’ on the union \(\overline T\cup\partial T\), and it is shown here that this union, provided with the observers’ topology, is a compact space \(\widehat T^{\text{obs}}\).

To any \(\mathbb{R}\)-tree \(T\) as above a ‘dual lamination’ \(L^2(T)\) has been associated by the authors [in J. Lond. Math. Soc., II. Ser. 78, No. 3, 737-754 (2008; Zbl 1198.20023)]. Here we prove that, if two such trees \(T_0\) and \(T_1\) have the same dual lamination \(L^2(T_0)=L^2(T_1)\), then with respect to the observers’ topology the two trees have homeomorphic compactifications: \(\widehat T^{\text{obs}}_0=\widehat T^{\text{obs}}_1\). Furthermore, if both \(T_0\) and \(T_1\), say with metrics \(d_0\) and \(d_1\), respectively, are minimal, this homeomorphism restricts to an \(F_N\)-equivariant bijection \(T_0\to T_1\), so that on the identified set \(T_0=T_1\) one obtains a well defined family of metrics \(\lambda d_1+(1-\lambda)d_0\). We show that for all \(\lambda\in[0,1]\) the resulting metric space \(T_\lambda\) is an \(\mathbb{R}\)-tree.

To any \(\mathbb{R}\)-tree \(T\) as above a ‘dual lamination’ \(L^2(T)\) has been associated by the authors [in J. Lond. Math. Soc., II. Ser. 78, No. 3, 737-754 (2008; Zbl 1198.20023)]. Here we prove that, if two such trees \(T_0\) and \(T_1\) have the same dual lamination \(L^2(T_0)=L^2(T_1)\), then with respect to the observers’ topology the two trees have homeomorphic compactifications: \(\widehat T^{\text{obs}}_0=\widehat T^{\text{obs}}_1\). Furthermore, if both \(T_0\) and \(T_1\), say with metrics \(d_0\) and \(d_1\), respectively, are minimal, this homeomorphism restricts to an \(F_N\)-equivariant bijection \(T_0\to T_1\), so that on the identified set \(T_0=T_1\) one obtains a well defined family of metrics \(\lambda d_1+(1-\lambda)d_0\). We show that for all \(\lambda\in[0,1]\) the resulting metric space \(T_\lambda\) is an \(\mathbb{R}\)-tree.

##### MSC:

20E08 | Groups acting on trees |

20E05 | Free nonabelian groups |

20F65 | Geometric group theory |

20F69 | Asymptotic properties of groups |

37A25 | Ergodicity, mixing, rates of mixing |

57M07 | Topological methods in group theory |

37E25 | Dynamical systems involving maps of trees and graphs |