Simple Groups Acting on Nonspherical Phan Geometries

Definitions and mathematical statement

For a building or twin building of Coxeter type $(W,S)$, the type is called nonspherical when the Coxeter group $W$ is infinite. A group $H$ is simple if $H\neq\{1\}$ and its only normal subgroups are $\{1\}$ and $H$.

The supplied source asks about groups denoted by $G_\sigma$ acting on Phan geometries associated with good flips $\sigma$. It does not define the ambient group $G$, the admissible buildings, a good flip, the associated Phan geometry, or the construction of $G_\sigma$. The supplied background describes the question as a centralizer question. If $\sigma$ is an involutive automorphism of $G$, its fixed subgroup is \[ G^\sigma=\{g\in G:\sigma(g)=g\}, \] which can also be identified with the centralizer of $\sigma$ in the copy of $G$ inside $G\rtimes\langle\sigma\rangle$. Identifying the source notation $G_\sigma$ with this subgroup requires confirmation from the report; the record does not exclude a quotient, an action image, or a related subgroup.

Problem (source formulation, with unresolved scope). In the nonspherical setting intended by the source, determine the groups $G_\sigma$ acting on the Phan geometries associated with good flips $\sigma$. In particular, determine whether these groups are simple. If the simplicity question is intended universally over the admissible pairs $(G,\sigma)$, its assertion would be \[ \text{for every admissible }(G,\sigma),\qquad G_\sigma\neq\{1\},\qquad \left(N\trianglelefteq G_\sigma\Longrightarrow N=\{1\}\ \text{or}\ N=G_\sigma\right). \] The admissible pairs and the precise meaning of $G_\sigma$ remain unspecified. The record also does not state whether simplicity is abstract or topological, or whether a central quotient is intended. Thus this is a faithful schematic statement requiring source clarification before a fully quantified formulation is possible.

Short English statement

Determine the groups acting on nonspherical Phan geometries arising from good flips, and decide whether they are simple. The ambient groups, good-flip hypotheses, acting-group construction, and intended notion of simplicity need clarification from the source.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30001109$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Repository registry locations: data/unsolvedmath/problems.json and supplementary catalog/problems.json. No individual proposer is identified in the supplied record.
• Supplied report citation: Mini-Workshop: Symmetric Varieties and Involutions of Algebraic Groups (2009), Oberwolfach Reports, pp. 2985--3028, DOI https://doi.org/10.4171/owr/2008/53. The record separately assigns proposed year 2008 and the tag owr-year:2008; the relationship between these dates is unresolved.
• Supplied literature source, labeled "Horn involutions of Kac--Moody groups": https://doi.org/10.26083/tuprints-00001900.
• Supplied literature source, labeled "Carbone--Ershov--Ritter": https://doi.org/10.1016/j.jpaa.2008.03.023.
• Supplied literature source, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2008/53.
• Literature assessment checked at 2026-08-21: the supplied dated curation labels the problem partially solved and states that simplicity is established for many Kac--Moody completions and numerous classical Phan cases, while the general nonspherical good-flip centralizer question remains unresolved. This is the supplied assessment, not an independently verified mathematical or literature conclusion. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with the explanation that the statement received meaning-preserving semantic, typographic, and LaTeX cleanup without unsupported source-dependent additions. This records the supplied editorial qualification; it does not establish mathematical completeness or independent source verification.