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Math-1787 · mathematics · Group theory

Questions in Geometric Group Theory — Q 1.6

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Statement

Definitions and mathematical statement

For a finite symmetric generating set $S$ of a group $G$, its Cayley graph has vertices $G$ and unit-length edges from $g$ to $gs$ for $s\in S$. The associated path metric is the word metric. A finitely generated group is word-hyperbolic if some $\delta\geq0$ makes every geodesic triangle in this graph $\delta$-thin: each side lies in the $\delta$-neighborhood of the other two. This property is independent of the finite generating set.

The group is one-ended if its Cayley graph is infinite and deleting any finite set of vertices leaves exactly one infinite connected component. Let $\Sigma_g$ denote the closed connected orientable surface of genus $g$. Its fundamental group has presentation \[ \pi_1(\Sigma_g)=\langle a_1,b_1,\ldots,a_g,b_g\mid [a_1,b_1]\cdots[a_g,b_g]=1\rangle, \qquad [a,b]=aba^{-1}b^{-1}. \] Does every one-ended word-hyperbolic group $G$ admit an injective homomorphism $\pi_1(\Sigma_g)\to G$ for some integer $g\geq2$?

The genus may depend on $G$, and no bound on it is requested. The surface subgroup is not required to have finite index or to be quasiconvex. Allowing nonorientable closed hyperbolic surfaces gives the same existence question, since their orientation double covers supply orientable surface subgroups. The assumption of one end excludes free groups, whose subgroups are free and therefore cannot be closed hyperbolic surface groups. This is a subgroup-existence question, rather than the construction of a quotient onto a surface group.

Short English statement

Must every one-ended hyperbolic group contain the fundamental group of a closed surface of genus at least two?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1787-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1787

The main theorem and its closed-hyperbolic-three-manifold consequences settle the S^2-boundary subcase. The record quantifies over additional boundary types, general hyperbolic groups, dimensions, or independent bundle clauses that this paper does not settle. This entry records partial support; complete coverage of the question is not established by this catalogue.

Questions in Geometric Group Theory — Q 1.6

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. A hyperbolic group with two-sphere boundary is claimed to act geometrically on hyperbolic three-space with finite kernel.

Mathematical statement of the claimed result

Result 1.
Let $G$ be word-hyperbolic with Gromov boundary $\partial G\cong S^2$. The source claims a homomorphism \[ \rho:G\longrightarrow\operatorname{Isom}(\mathbb H^3) \] whose action on hyperbolic three-space is proper and cocompact, and whose kernel is finite. With the additional assumption that $G$ is torsion-free, the corollaries identify $G\cong\pi_1(M)$, where $M$ is closed and has a hyperbolic metric in dimension three, and supply a finite-index subgroup \[ H\cong\pi_1(\Sigma)\rtimes\mathbb Z, \] where $\Sigma$ is a closed connected orientable surface. They also assert residual finiteness, virtual compact specialness, and positive virtual first Betti number in that torsion-free case.

Outline of the argument

The argument applies a boundary modulus criterion. Unbounded modulus produces scalar limits analyzed by exponential records or subexponential probability flows; comparing same-depth expansions forces the required measure control and the three-dimensional hyperbolic action.

Formal verification

For family 246, the documented theorem gives a proper cocompact hyperbolic-three-space action with finite kernel from the two-sphere boundary hypothesis. The listed Comparator files are CannonGeometricAction.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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