Virtual Simplicity of Exotic Affine-Building Lattices

Definitions and mathematical statement

An affine building of type $\widetilde A_2$ is a Euclidean building whose apartments have the triangular Coxeter tessellation of type $\widetilde A_2$. A Bruhat--Tits building means an affine building arising from the Bruhat--Tits construction for an algebraic group over a nonarchimedean valued field; the condition below concerns isomorphism as buildings.

A group $H$ is simple if $H\ne\{1\}$ and its only normal subgroups are $\{1\}$ and $H$. A group $\Gamma$ is virtually simple if it contains a simple subgroup of finite index, that is, \[ \exists H\leq\Gamma\quad [\Gamma:H]<\infty,\qquad H\ne\{1\},\qquad \forall N\trianglelefteq H\quad (N=\{1\}\ \lor\ N=H). \] The subgroup $H$ is not required to be normal in $\Gamma$.

The supplied record does not define an $\widetilde A_2$-lattice. Under the usual topological interpretation, when $X$ is locally finite, $\operatorname{Aut}(X)$ has the compact-open topology, and a lattice is a discrete subgroup $\Gamma\leq\operatorname{Aut}(X)$ for which $\operatorname{Aut}(X)/\Gamma$ carries a nonzero finite invariant measure. Cocompactness additionally means that this quotient is compact. The record does not specify whether this interpretation is intended, whether cocompactness is required, or whether a narrower class of actions is intended. These choices are therefore left unresolved.

Conjecture as supplied, with its unresolved domain made explicit. Let $\mathcal L_{\mathrm{source}}$ denote the intended class of pairs $(X,\Gamma)$ consisting of an associated building and an $\widetilde A_2$-lattice acting on it. Is it true that \[ \forall (X,\Gamma)\in\mathcal L_{\mathrm{source}},\qquad X\text{ is not a Bruhat--Tits building} \ \Longrightarrow\ \exists H\leq\Gamma\; \left([\Gamma:H]<\infty\ \land\ H\text{ is simple}\right)? \] This records the universal conjecture without resolving the missing definition of $\mathcal L_{\mathrm{source}}$. In particular, the reference to cocompact lattices in the supplied literature assessment is not silently imposed on the conjecture.

Short English statement

The conjecture asks whether every $\widetilde A_2$-lattice whose associated building is not Bruhat--Tits contains a simple subgroup of finite index. The intended class of lattices, including any cocompactness or local finiteness requirements, needs clarification. The supplied partially solved label is dated curation, not an independently verified conclusion.

Sources

• Local problem registry: data/unsolvedmath/problems.json, with catalog/problems.json as the supplementary registry; permanent UnsolvedMath ID $30004455$, title "Virtual Simplicity of Exotic Affine-Building Lattices," source code catalogue entry. This draft uses the supplied complete registry record; the registry files were not independently inspected.
• Supplied report citation: Geometric Structures in Group Theory (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/16, pp. 877--918. https://doi.org/10.4171/owr/2020/16. The record attributes the problem to Oberwolfach workshop discussions, gives proposed year 2020, and supplies no individual proposer or exact problem page.
• Supplied literature source: Bader--Furman--Lecureux (2025), https://arxiv.org/abs/2310.13472.
• Supplied literature source: Lecureux--Witzel (2026), https://arxiv.org/abs/2605.06163.
• Supplied literature source: Oberwolfach Report 16/2020, https://doi.org/10.4171/owr/2020/16.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. Its status label is partially_solved. It reports that normal-subgroup theorems show that cocompact lattices on $\widetilde A_2$ buildings have finite-index nontrivial normal subgroups, describes the virtual-simplicity conjecture as reduced to residual-finiteness issues, and says the conjecture for every non-Bruhat--Tits lattice has not been proved in full. The record explicitly characterizes this as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims. None of these references or claims was independently checked for this draft.