Uncountably many homogeneous real trees with the same valence

TL;DR

The paper addresses the question of whether valence alone classifies homogeneous real trees in the incomplete setting. It constructs a filtration of the universal real tree $T_\kappa$ by Cantor-Bendixson complexity, producing incomplete homogeneous $\mathbb{R}$-trees $T_\kappa^{[\alpha]}$ of valence $\kappa$ and showing the directed union is isometric to $T_\kappa$. The main result is that there exist uncountably many pairwise non-isometric such trees for each $\kappa \ge 3$, highlighting that valence does not determine isometry class without completeness. The work uses Cantor-Bendixson ranks, explicit isometry constructions, and filtration techniques to connect incomplete homogeneous trees to the universal tree, with implications for geometric group theory and asymptotic cone analysis.

Abstract

For any cardinal $κ\geq 2$, there is a unique complete real tree whose points all have valence $κ$. In this note, we show that, when $κ\geq 3$, it is necessary to assume completeness. More precisely, we show that there exist uncountably many homogeneous incomplete real trees whose points all have valence $κ$.

Theorems & Definitions (31)

• Theorem 1.1
• proof
• Remark 1.2
• Remark 2.1
• Definition 3.1: Labels
• Definition 3.2
• Definition 3.3: Partial order
• Remark 3.4
• Definition 3.5: Metric
• Definition 3.6: Complexity