Nontrivial Floyd Boundaries Beyond Relative Hyperbolicity

Definitions and mathematical statement

For a finitely generated group $G$ with a finite symmetric generating set, let $d$ be its Cayley-graph distance from the identity. A standard Floyd construction uses a positive summable function $f:{\mathbb{N}}\to(0,\infty)$ and gives an edge $\{u,v\}$ length $f(\min\{d(1,u),d(1,v)\})$. The induced path metric on vertices has a completion $\overline G_f$, and the Floyd boundary is $\partial_fG=\overline G_f\setminus G$. Standard regularity conditions often include monotonicity and a uniform lower bound on $f(n+1)/f(n)$; the source instead refers to its own unstated standard conditions, which must be recovered.

Retain all three questions from the source:

1. For the intended allowed Floyd function, does there exist $G$ with nontrivial $\partial_fG$ that is not relatively hyperbolic with respect to any allowed family of finitely generated peripheral subgroups?
2. For the particular infinitely presented group $H$ constructed in the report, is $\partial_fH$ trivial?
3. Is that same $H$ not relatively hyperbolic with respect to every such family?

The second and third are separate questions about a specified but unrecovered group; no answers are inferred from its description as a candidate.

For a substantive relative-hyperbolicity question, peripheral structures must exclude the vacuous choice $\{G\}$; the usual setting uses proper peripheral subgroups. The extract does not specify whether the family is finite or which degeneracies are permitted. Likewise "nontrivial Floyd boundary" may mean at least three points in the conventional nonelementary sense, rather than merely nonempty; the exact cardinality convention is absent.

The original keeps $f$ subject to the report's conditions, while the clean statement suppresses it as a hypothesis. Its quantifier (fixed, arbitrary allowed, or existential) and the finite-generation/generating-set conventions must therefore be confirmed. The group $H$ has no presentation or construction in this record and is not replaced by another group mentioned in the workshop.

Short English statement

Seek a nontrivial Floyd boundary without any permissible relatively hyperbolic structure, and decide two separate boundary/hyperbolicity questions for the report's infinitely presented group. The Floyd hypotheses, peripheral conventions and candidate group are not supplied.

Sources

• Supplied report: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406; proposed year 2008. https://doi.org/10.4171/owr/2008/42. The verification identifies Questions 6--8.
• Supplied Yang reference on peripheral structures: https://arxiv.org/abs/1101.2990.
• Supplied Mahan reference on controlled Floyd separation: https://mathweb.tifr.res.in/~mahan/floyd.pdf.
• Local UnsolvedMath registry, permanent ID 30001036, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note says the general existence and both candidate questions are preserved. Literature checked at 2026-08-21: the assessment is supplied dated curation, not independently verified. It labels the general nontrivial-Floyd-implies-relative-hyperbolicity conjecture unresolved while mentioning counterexamples to weaker formulations; their precise versions are not provided.