Minimal Fold of Kazhdan Hyperbolic Triangle Groups

Definitions and mathematical statement

Let $K\subseteq{\mathbb{Z}}$ denote the admissible fold indices, and for each $k\in K$ let $\mathcal C_k$ denote the class called generalized $k$-fold triangle groups in the cited sources. The supplied record specifies neither $K$ nor the construction or defining conditions of $\mathcal C_k$. These symbols therefore stand for unresolved source-dependent data, not definitions of that class.

Under the usual word-hyperbolic interpretation, a group $\Gamma$ is hyperbolic if it has a finite generating set whose Cayley graph, with edges of length one, has $\delta$-thin geodesic triangles for some $\delta\geq0$: every side lies in the closed $\delta$-neighborhood of the union of the other two sides. For a discrete group $\Gamma$, Kazhdan property $(T)$ means that every unitary representation $\pi:\Gamma\to\mathcal U(H)$ on a complex Hilbert space having almost invariant unit vectors has a nonzero invariant vector. Here almost invariance means that for every finite $F\subseteq\Gamma$ and every $\varepsilon>0$, there is a unit vector $v\in H$ such that $\|\pi(g)v-v\|<\varepsilon$ for all $g\in F$; invariance means that some $w\neq0$ satisfies $\pi(g)w=w$ for every $g\in\Gamma$.

The requested target is to determine \[ k_{\min}=\min\left\{k\in K:\ \exists\Gamma\in\mathcal C_k\text{ such that }\Gamma\text{ is word-hyperbolic and has property }(T)\right\}. \] This formulation remains conditional on recovering the intended class and admissible indices. The record does not explicitly require infinitude or non-elementarity; whether either condition belongs to the intended class must be checked in the supplied references. The supplied literature assessment reports $3\leq k_{\min}\leq5$ and labels the minimum unresolved. These are supplied curation claims, not independently verified conclusions.

Short English statement

Determine the smallest admissible fold index for which a generalized triangle group is hyperbolic and has Kazhdan property $(T)$. The defining generalized triangle-group construction is missing from the supplied record, so a fully self-contained statement requires source review. Supplied dated curation reports that the minimum lies between $3$ and $5$ and remains open.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004459$, title: Minimal Fold of Kazhdan Hyperbolic Triangle Groups; source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified.
• Supplied source citation: Geometric Structures in Group Theory (2021), Oberwolfach Reports, pp. 877--918, DOI 10.4171/owr/2020/16. https://doi.org/10.4171/owr/2020/16. The record assigns proposed year 2020.
• Supplied literature reference, labeled "generalized triangle groups paper": https://doi.org/10.1112/jlms.12668. No authors, title, or relevant definition or theorem location are supplied for this reference.
• Literature checked at: 2026-08-22. The supplied dated curation labels the problem open and states that the primary construction establishes only the bounds $3\leq k\leq5$, leaving the smallest fold unresolved. This assessment is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.