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Non-unique ergodicity, observers’ topology and the dual algebraic lamination for \(\mathbb{R}\)-trees. (English) Zbl 1197.20020
Summary: 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.

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
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