Complex Hyperbolic Plane Embeddings into Four Trees

Definitions and mathematical statement

Let $\mathbb H^2_{{\mathbb{C}}}$ be the complex hyperbolic plane, of complex dimension two and real dimension four, with its standard complex-hyperbolic distance $d_{\mathrm{CH}}$. A constant rescaling of that metric does not change the quasi-isometric embedding question. A metric tree is understood here in the geodesic-tree sense: between two points there is a unique arc, isometric to an interval. This includes metric realizations of simplicial trees.

The question is whether there exist four metric trees $(T_i,d_i)$, a map \[ F:\mathbb H^2_{{\mathbb{C}}}\longrightarrow T_1\times T_2\times T_3\times T_4, \] and constants $L\ge1$, $C\ge0$ such that for every $x,y\in\mathbb H^2_{{\mathbb{C}}}$, \[ L^{-1}d_{\mathrm{CH}}(x,y)-C \le\sum_{i=1}^4d_i(F_i(x),F_i(y)) \le Ld_{\mathrm{CH}}(x,y)+C. \] The displayed product uses the sum metric. Other fixed $\ell^p$ product metrics are bi-Lipschitz equivalent for four factors, so this choice does not alter existence. No coarse-surjectivity condition is part of quasi-isometric embedding. The source does not require the trees to be locally finite, of bounded valence, or simplicial; if those restrictions are intended, they must be checked rather than added.

The literature assessment checked on 21 August 2026 says the cited tree-product results give nearby bounds but do not settle the four-tree case. No alternative number of factors is substituted for the exact target. This open-status assessment is supplied dated triage requiring independent verification.

Short English statement

Can the complex hyperbolic plane be embedded with uniformly controlled large-scale distances into a product of four trees? The question requires four factors and no coarse-surjectivity condition.

Sources

• Geometrie (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/47, pp. 2491--2538, https://doi.org/10.4171/owr/2004/47; proposed year supplied as 2004.
• Literature assessment checked 21 August 2026: original report, https://doi.org/10.4171/owr/2004/47; "embedding in products of trees," https://arxiv.org/abs/math/0311524.
• Local UnsolvedMath registry, permanent ID $30000138$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. Corrected-and-verified status records removal of a redundant label and balanced projective/hyperbolic notation.