The Boundary Singer Conjecture

Definitions and mathematical statement

Under the usual interpretation of the notation, a finite $d$-dimensional classifying complex $B\Gamma$ is a finite connected CW complex of dimension $d$ with fundamental group $\Gamma$ and contractible universal cover. A group $\Gamma$ is residually finite if, for every $g\in\Gamma\setminus\{1\}$, there is a homomorphism to a finite group whose value on $g$ is nonidentity. Let ${\mathbb{F}}_2={\mathbb{Z}}/2{\mathbb{Z}}$, and let $S^n$ denote the topological $n$-sphere for $n\geq 0$.

Assume that $\Gamma$ is residually finite and is a CAT(0) group or a hyperbolic group, and that it admits such a complex $B\Gamma$. In the hyperbolic case, the usual interpretation of $\partial\Gamma$ is the Gromov boundary. In the CAT(0) case, the usual interpretation is the visual boundary of a proper CAT(0) space on which $\Gamma$ acts properly and cocompactly by isometries; the supplied record does not specify this space or quantify over its choices. Write \[ V=H^{\mathrm{st}}_{d-1}(\partial\Gamma;{\mathbb{F}}_2) \] for the boundary homology group appearing in the source. The supplied record does not expand the superscript $\mathrm{st}$, specify the homology convention, or define the entropy of the induced action $\Gamma\curvearrowright V$.

Boundary Singer Conjecture, as supplied. For every group and classifying complex satisfying the preceding hypotheses, is the following implication true? \[ \text{The induced action }\Gamma\curvearrowright V \text{ has positive entropy} \quad\Longrightarrow\quad \nexists\,\text{topological embedding } \partial\Gamma\hookrightarrow S^{2d-2}. \] Here a topological embedding is a continuous map that is a homeomorphism onto its image. With ordinary nonnegative-dimensional sphere notation, this formulation requires $d\geq 1$; the supplied record does not state a lower bound on $d$ or a convention for a negative-dimensional target sphere.

The implication is conjectural, not an asserted theorem. Its fully precise interpretation remains conditional on specifying the boundary in the CAT(0) case, the homology theory, and the entropy invariant, including any auxiliary choices and their quantifiers.

Short English statement

The supplied Boundary Singer Conjecture asks whether positive entropy of the group action on the indicated boundary homology forces the boundary to have no topological embedding into $S^{2d-2}$. The group must be residually finite, CAT(0) or hyperbolic, and admit a finite $d$-dimensional classifying complex. The boundary choice, homology convention, entropy definition, and dimension range require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004387$, title The Boundary Singer Conjecture, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the supplied proposed year is 2020.
• Supplied source citation: Manifolds and Groups (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/7, pp. 417--463. https://doi.org/10.4171/owr/2020/7. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: "Avramidi report on Singer." https://doi.org/10.4171/owr/2020/7.
• Supplied literature reference: "Avramidi--Okun--Schreve 2025." https://doi.org/10.1007/s00208-025-03164-z.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and states that Singer's concentration conjecture remains open in general, with proofs for substantial classes including many hyperbolic, cubical, and branched-cover cases. This is supplied dated curation, not independently verified here. The record itself requests independent verification of time-sensitive status and cited claims; it does not specify which results establish cases of this exact boundary formulation.
• Supplied statement-verification provenance: the record labels the statement corrected_verified and reports that the official report was checked, that it calls this the Boundary Singer Conjecture, and that a declarative extraction was consequently restored to a conjectural question. This records the supplied verification claim and does not assert an independent source visit.