Boundary and Hyperbolic Dimensions of Symmetric Spaces

Definitions and mathematical statement

Let $\Gamma$ be a finitely generated Gromov-hyperbolic group, equipped with a word metric, and let $\partial_\infty\Gamma$ be its boundary. For a basepoint $o$, a visual metric has the form \[ C^{-1}a^{-(\xi|\eta)_o}\leq d_v(\xi,\eta) \leq Ca^{-(\xi|\eta)_o}\qquad(\xi\ne\eta) \] for suitable $a>1$, $C\geq1$ and the extended Gromov product. Write $\dim$ for topological covering dimension. The record asks three separate questions:

1. Is $\dim\partial_\infty\Gamma=\operatorname{mdim}(\partial_\infty\Gamma,d_v)$ for a visual metric?
2. What is the exact value of $\operatorname{hdim}(\mathbb H^2\times\mathbb H^2)$, with the usual product metric on the two hyperbolic planes?
3. For every noncompact symmetric space $X$ without Euclidean factors, is $\operatorname{hdim}(X)\geq\dim X$?

Here $\operatorname{hdim}$ is the report's hyperbolic dimension, not Hausdorff dimension, and $\operatorname{mdim}$ is its boundary metric-dimension invariant. The supplied extraction does not define either invariant. In particular, the latter cannot be silently identified with Hausdorff dimension, mean dimension, capacity dimension, or an Assouad-type dimension: their definitions and metric dependence differ. A source-specific cover definition is needed to make the first and the hyperbolic-dimension questions fully precise. The boundary degeneracies for elementary hyperbolic groups and the choice of visual metric parameters are also not discussed.

For comparison, asymptotic dimension $\operatorname{asdim}X\leq n$ means that for every scale $R>0$ there is a uniformly bounded cover with $R$-multiplicity at most $n+1$. The supplied assessment reports the Gromov asymptotic-dimension question from the passage as proved by Buyalo and Lebedeva; the usual associated hyperbolic-group formula is $\operatorname{asdim}\Gamma=\dim\partial_\infty\Gamma+1$ in its non-elementary setting. This reported result must be checked against the missing definition of $\operatorname{mdim}$ before identifying it with the exact first displayed equality.

The third question also needs the intended symmetric-space convention: "of noncompact type" normally excludes compact factors as well as Euclidean ones, whereas literal noncompactness alone does not exclude compact factors. The source does not spell out this distinction or the covering/connectedness convention. No additional hypothesis is silently inserted to settle the question.

The supplied literature assessment checked 21 August 2026 labels the record partially solved: it reports the asymptotic-dimension progress while saying the exact hyperbolic dimension of $\mathbb H^2\times\mathbb H^2$ remains unresolved in the located literature. It gives no exact value or resolution of the full symmetric-space inequality. These are dated supplied claims, not an independent current-status survey.

Short English statement

The entry combines a boundary metric-dimension equality, the exact hyperbolic dimension of two hyperbolic planes, and a proposed lower bound for symmetric spaces. The supplied assessment reports asymptotic-dimension progress but not the exact product value. Missing definitions of the two specialized dimensions and the symmetric-space convention require source review.

Sources

• Local UnsolvedMath registry, permanent ID 30000139, exact source code catalogue entry; Boundary and Hyperbolic Dimensions of Symmetric Spaces. Proposed year 2004, status partially_solved; corrected_verified extraction retains all three questions and repairs infinity subscripts.
• Supplied report citation: Geometrie (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/47, pp. 2491--2538, https://doi.org/10.4171/owr/2004/47. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21: "original report", https://doi.org/10.4171/owr/2004/47; "Buyalo--Lebedeva theorem", https://arxiv.org/abs/math/0509433; "products of hyperbolic spaces", https://arxiv.org/abs/math/0609463. This dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.