4.3 (Cooper) — 3-manifold groups acting on the affine building for $SL(4,\mathbb{R})$

Definitions and mathematical statement

Question (Cooper). For which closed $3$-manifolds $M$ does $\pi_1(M)$ act on the affine building referred to in the source as "the affine building for $\mathrm{SL}(4,{\mathbb{R}})$," with quotient retracting to $M$?

If $X$ denotes the intended building and $G=\pi_1(M)$, the requested condition is an action \[ \rho:G\longrightarrow\operatorname{Aut}(X) \] for which $Y=\rho(G)\backslash X$ contains a copy $i(M)$ of $M$ and admits a continuous retraction \[ r:Y\longrightarrow M,\qquad r\circ i=\operatorname{id}_M. \] This uses "retracts" in its ordinary topological sense. A deformation retraction is not asserted by the excerpt. Nor does the excerpt require the action to be faithful, free, proper, or cocompact; those additional conditions cannot be supplied silently. The problem asks when this property holds, rather than explicitly asserting it for all closed $3$-manifolds.

The building $X$ is not defined in the supplied record. An affine-building construction over a field generally requires valuation or other specified limiting data. The notation $\mathrm{SL}(4,{\mathbb{R}})$ alone does not identify such data, and its usual real symmetric space is not to be silently substituted for an affine building. Likewise, replacing ${\mathbb{R}}$ by a non-Archimedean local field would change the stated problem. The precise model and automorphism/action conventions therefore remain essential unresolved definitions. The source gives $M=\operatorname{Vol}3$ as an example; this notation is retained literally because the record supplies no further identification.

Short English statement

Determine which closed $3$-manifolds have fundamental-group actions on the source's specified affine building whose quotient contains the manifold as a retract. The affine-building model for $\mathrm{SL}(4,{\mathbb{R}})$ and the intended action conditions need clarification.

Sources

• Local UnsolvedMath registry, permanent ID 10900022, source code catalogue entry; attributed to Cooper, no proposal year supplied.
• Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), subsection 4.3 (Cooper), PDF page 3; https://arxiv.org/pdf/1512.04620#page=3 and https://arxiv.org/abs/1512.04620. The provenance records source-TeX extraction checked against rendered PDF text and access on 2026-07-29; the source example is written $\operatorname{Vol}3$.
• The registry research summary discusses affine buildings, $p$-adic groups, and possible construction techniques but supplies no primary reference defining the building or resolving the retraction question.