Hyperbolic Rank of Affine Buildings

Definitions and mathematical statement

Equip an affine building $X$ with its standard Euclidean building metric $d_X$. A map $F:Y\to X$ is a quasi-isometric embedding if there are constants $A\geq1$, $B\geq0$ such that \[ A^{-1}d_Y(y,y')-B\leq d_X(F(y),F(y')) \leq A d_Y(y,y')+B\qquad(y,y'\in Y). \] No coarse-surjectivity condition is imposed. The source defines hyperbolic rank as the largest boundary dimension of a Gromov-hyperbolic space admitting such an embedding. With an explicitly chosen admissible class $\mathcal C$ of hyperbolic spaces, this is expressed as \[ \operatorname{rank}_h(X)=\sup\{\dim_{\mathrm{top}}\partial_\infty Y: Y\in\mathcal C,\ Y\text{ is Gromov-hyperbolic and quasi-isometrically embeds in }X\}. \] The record does not specify the properness, geodesicity, bounded-geometry, or boundary-topology conditions on $\mathcal C$, nor whether the supremum is attained. It asks to determine this invariant for affine buildings; it is not the dimension of a maximal Euclidean apartment.

The explicit model question is whether every intended affine building $X$ of type $\widetilde A_2$ admits a quasi-isometric embedding \[ F:\mathbb H^2\longrightarrow X. \] The constants and the map may depend on $X$. Since $\partial_\infty\mathbb H^2\cong S^1$, an affirmative answer would give the lower bound $\operatorname{rank}_h(X)\geq1$ when $\mathbb H^2\in\mathcal C$; the converse is not supplied by the definition alone.

Thickness is an essential missing convention. A thin building consisting of one $\widetilde A_2$ apartment is metrically the Euclidean plane. It cannot contain a quasi-isometrically embedded $\mathbb H^2$: large separated sets in hyperbolic balls grow exponentially, whereas separated sets in comparable Euclidean balls grow only polynomially. Thus if thin buildings are allowed, the universal model question has this immediate obstruction. The intended building class may require thickness, local finiteness, or other conditions; these must be recovered rather than silently imposed.

The supplied assessment dated 2026-08-21 reports a solved product-of-trees case but says the arbitrary intended $\widetilde A_2$-building question remains open, and that newer results do not give a single hyperbolic-plane embedding in every such building. It supplies no exact value for the product case or full hypotheses of the newer results.

Short English statement

The problem asks how large a hyperbolic boundary can be represented inside an affine building by a map preserving distances up to uniform multiplicative and additive errors. Its concrete case asks for one hyperbolic plane in each intended type-$\widetilde A_2$ building; the missing thickness convention is crucial.

Sources

• Local UnsolvedMath registry, permanent ID $30000912$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Buildings: Interactions with Algebra and Geometry (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/03, pp. 119--172. https://doi.org/10.4171/owr/2008/03.
• Supplied reference labelled "Schroeder report": https://ems.press/content/serial-article-files/46148.
• Supplied reference: Bensaid--Nguyen, https://arxiv.org/abs/2405.02226.
• The supplied corrected_verified note reports checking mathematical content against the official report. The open label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of scope and status, including the building conventions.