[Paper Review] Non-unique ergodicity, observers' topology and the dual algebraic lamination for $\R$-trees

Thierry Coulbois, Arnaud Hilion+1 more|arXiv (Cornell University)|Jun 9, 2007

Geometric and Algebraic Topology4 citations

TL;DR

This paper establishes that two $ R$-trees with very small, minimal, and dense-orbit actions of a free group $F_N$ are homeomorphic under the observers' topology if and only if they share the same dual algebraic lamination $L^2(T)$. Furthermore, when such trees are identified via this homeomorphism, any convex combination of their metrics yields a new $ R$-tree, proving non-unique ergodicity in the context of $ R$-trees via metric interpolation.

ABSTRACT

We continue in this article the study of laminations dual to very small actions of a free group F on R-trees. We prove that this lamination determines completely the combinatorial structure of the R-tree (the so-called observers' topology). On the contrary the metric is not determined by the lamination, and an R-tree may be equipped with different metrics which have the same observers' topology.

Motivation & Objective

• To establish a topological characterization of $ R$-trees with very small $F_N$-actions via the observers' topology.
• To investigate the relationship between the dual algebraic lamination $L^2(T)$ and the topological structure of $ R$-trees with dense orbits.
• To determine whether two $ R$-trees with the same dual lamination are topologically equivalent under the observers' topology.
• To explore the geometric consequences of non-unique ergodicity in $ R$-trees by constructing new $ R$-tree metrics via linear combinations of existing ones.
• To prove that the resulting metric space from such interpolation remains an $ R$-tree, preserving the necessary geometric and topological properties.

Proposed method

• Introduces the observers' topology on the union $ar{T} igcup d T$, where $ar{T}$ is the metric completion of an $ R$-tree $T$ and $d T$ is its Gromov boundary, showing this topology is compact and coarser than the metric topology.
• Uses the $F_N$-equivariant map ${rak Q}: d F_N o ar{T} igcup d T$ to define the dual algebraic lamination $L^2(T)$ as the set of pairs of distinct boundary points mapping to the same point under ${rak Q}$.
• Applies the equivalence of three definitions of $L^2(T)$ from prior work [CHLII], particularly the limit of conjugacy classes with vanishing translation length.
• Establishes that $L^2(T_0) = L^2(T_1)$ implies $F_N$-equivariant homeomorphism between the compactified spaces $ H{T}_0^{ ext{obs}}$ and $ H{T}_1^{ ext{obs}}$ via the observers' topology.
• Identifies the interiors of $T_0$ and $T_1$ as the same set $ T^ullet$, allowing the definition of a combined metric $d_ u = u d_1 + (1- u)d_0$ for $ u o [0,1]$.
• Proves that the resulting metric space $( T^ullet, d_ u)$ satisfies the Gromov $0$-hyperbolicity condition and admits unique geodesics, confirming it is an $ R$-tree.

Experimental results

Research questions

• RQ1 Under what topological conditions do two $ R$-trees with very small $F_N$-actions and dense orbits become homeomorphic?
• RQ2 Can the dual algebraic lamination $L^2(T)$ uniquely determine the topological structure of an $ R$-tree under the observers' topology?
• RQ3 Does the existence of multiple non-isometric metrics on the same $ R$-tree (arising from different measures) lead to a well-defined interpolation that preserves the $ R$-tree structure?
• RQ4 Is the linear combination of two metrics on the same $ R$-tree, derived from different $F_N$-actions, guaranteed to yield a new $ R$-tree?
• RQ5 How does the observers' topology relate to the metric topology on $ R$-trees, especially in the presence of infinite branching or dense orbit structures?

Key findings

• Two $ R$-trees $T_0$ and $T_1$ with very small, minimal, and dense-orbit $F_N$-actions are $F_N$-equivariantly homeomorphic under the observers' topology if and only if their dual algebraic laminations satisfy $L^2(T_0) = L^2(T_1)$.
• The $F_N$-equivariant homeomorphism between $ H{T}_0^{ ext{obs}}$ and $ H{T}_1^{ ext{obs}}$ restricts to a bijection between the interiors $T_0$ and $T_1$, identifying them as the same set $ T^ullet$.
• For any $ u o [0,1]$, the metric $d_ u = u d_1 + (1- u)d_0$ on $ T^ullet$ defines a new metric space that is an $ R$-tree.
• The geodesics in $( T^ullet, d_ u)$ coincide with those in $T_0$ and $T_1$, and the topology induced by $d_ u$ agrees with the observers' topology on the interior.
• The Gromov product condition for $0$-hyperbolicity is preserved under convex combination of metrics, ensuring the resulting space is $0$-hyperbolic and thus an $ R$-tree.
• The canonical map ${rak Q}$ from $d F_N$ to $ H{T}^{ ext{obs}}$ commutes with the $F_N$-equivariant homeomorphism, confirming consistency of the dual lamination as a topological invariant.