Boundaries of Groups and Kleinian Groups — Problem 86

Definitions and mathematical statement

A group $G$ is coherent if \[ \forall H\leq G,\qquad H\text{ finitely generated}\ \Longrightarrow\ H\text{ finitely presented}. \] This is the explicit algebraic property recalled in the source. It is a condition on every finitely generated subgroup, not merely on $G$ itself.

The actual problem is the broader request to find new restrictions on Kleinian groups. In a real-hyperbolic interpretation, these are discrete groups of isometries of $\mathbb H^n$; the classical three-dimensional usage concerns discrete subgroups of $\operatorname{PSL}_2({\mathbb{C}})$. The source does not specify the ambient dimension, torsion assumptions, finite generation, or geometric finiteness conditions.

No particular new restriction is proposed, and the recalled coherence definition is not itself a conjecture that all higher-dimensional Kleinian groups are coherent. The supplied summary suggests higher-dimensional questions after three-dimensional progress, but does not determine a single class or quantified assertion. A precise theorem target must first identify both the Kleinian-group class and the proposed algebraic or geometric restriction.

Short English statement

Find additional algebraic or geometric constraints on Kleinian groups. Coherence is supplied as context, without a conjecture asserting it for all dimensions.

Sources

• Local UnsolvedMath registry, permanent ID 6200086, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 86"; proposed year 2005, no proposer supplied. Registry status open, classification OPEN-TRIAGE.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 86, PDF page 22, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026. No PDF was accessed here.
• The supplied summary calls higher-dimensional restrictions open and describes three-dimensional coherence as settled, using the label "Scott conjecture" and an Agol attribution. It gives no precise theorem or reference; the relationship among compact-core/coherence and tameness statements and their attributions requires checking, so no exact historical theorem is inferred from that sentence.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the source is publicly author-hosted.