Realizing Convex Building Subcomplexes by Destabilizing Loci

Definitions and mathematical statement

Let $X$ be the building in the supplied question, and let $Y\subseteq X$ be a convex contractible subcomplex. Contractibility means that the geometric realization $|Y|$ admits a homotopy from its identity map to a constant map. The supplied record does not specify the class of buildings, the geometry used for convexity, or the group $G$ and its relationship to $X$.

Read stabilization as setwise stabilization. For the intended automorphism group $\operatorname{Aut}(X)$, define \[ \operatorname{Stab}_{\operatorname{Aut}(X)}(Y) =\{h\in\operatorname{Aut}(X):h(Y)=Y\}. \] The question is whether, for every such $Y$, there exist a variety $V$, an action $\alpha:G\times V\to V$, and a point $v\in V$ such that \[ C_0(v)\subseteq Y \quad\text{and}\quad h\left(C_0(v)\right)=C_0(v) \quad\text{for every }h\in\operatorname{Stab}_{\operatorname{Aut}(X)}(Y). \] Here $C_0(v)$ is retained as an undefined source symbol: the record supplies no construction of this subset of $X$, no definition of the admissible actions, and no base field or hypotheses on $G$ or $V$. It also does not state whether $C_0(v)$ must be nonempty. These omissions prevent a fully specified mathematical formulation; no definition of a destabilizing locus is being inferred.

The requested relation is containment $C_0(v)\subseteq Y$, together with invariance under the stabilizer of $Y$. The supplied literature assessment instead discusses realizing every such $Y$ as $C_0(v)$, which suggests equality. These formulations are not interchangeable, and the containment formulation above follows the supplied original and cleaned statements.

Short English statement

For a convex contractible subcomplex $Y$ of a building $X$, ask whether an action of the intended group $G$ on some variety and a point $v$ can produce a subset $C_0(v)\subseteq Y$ preserved by every automorphism preserving $Y$. The record leaves $G$, $C_0(v)$, and the admissible geometric and algebraic settings undefined, so this is a statement-only draft requiring clarification and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30001105$, source problem code catalogue entry, title: "Realizing Convex Building Subcomplexes by Destabilizing Loci." The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is $2008$.
• Supplied report citation: Mini-Workshop: Symmetric Varieties and Involutions of Algebraic Groups (2009), Oberwolfach Reports, pp. 2985--3028, DOI 10.4171/owr/2008/53. https://doi.org/10.4171/owr/2008/53. This is also the supplied literature link labeled "original Oberwolfach report." The citation's year $2009$ and the record's proposed year $2008$ are retained without reconciliation.
• Supplied literature reference labeled "Bate--Martin--Röhrle--Tange": https://doi.org/10.1090/S0002-9947-2012-05739-4. No title or further bibliographic details are supplied.
• Supplied literature reference labeled "fixed-point subcomplexes": https://arxiv.org/abs/0811.4294. No authors or further bibliographic details are supplied.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified. The record labels the problem open and reports that optimal-destabilizing-parabolic constructions answer it for fixed-point and other special subcomplexes, while no general construction realizing every convex contractible subcomplex as a $C_0(v)$ has been established. The background explicitly calls this dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims. Its equality language differs from the containment requested in the statement.
• The supplied statement status is corrected_verified, qualified by the record as meaning-preserving editorial and LaTeX cleanup with no source-dependent content added. This label and the supplied open status are registry assertions, not independent mathematical or source verification here.