Problem 6.1 — Are the Leininger–Reid subgroups geometrically finite, with cusp groups the reducible cyclic subgroups?

Definitions and mathematical statement

Problem 6.1 asks whether the specific Leininger--Reid subgroups are geometrically finite with cusp groups equal to the reducible cyclic subgroups. The record supplies neither the subgroup construction nor the definition of geometric finiteness in the mapping-class-group setting.

Conventionally a reducible mapping class preserves an essential multicurve, with periodic-versus-reducible conventions requiring specification. An infinite cyclic subgroup generated by such an element is a candidate peripheral group. However, "the reducible cyclic subgroups" does not state whether one means maximal such cyclic subgroups, their conjugacy classes, or all their subgroups. A cusp stabilizer and an arbitrary finite-index subgroup of it are not automatically the same designated peripheral object.

A convergence-action formulation of geometric finiteness would need a specified boundary action and definitions of conical and bounded-parabolic points; a relative-hyperbolicity formulation would need a peripheral family. The source does not select either formulation or give a geometric action from which cusp groups can be read. Thus the requested statement is the simultaneous geometric-finiteness and exact-peripheral-identification question for the source's particular groups, not an assertion about all surface subgroups, and not an unspecified Kleinian geometric-finiteness problem.

Short English statement

Are the named Leininger--Reid subgroups geometrically finite, with exactly the intended reducible cyclic cusp groups? Their construction, geometric-finiteness model and cyclic-peripheral maximality convention are absent from this extract.

Sources

• Local UnsolvedMath registry, permanent ID $11000241$, exact source code catalogue entry, Problem 6.1, PDF page 277. No individual proposer or proposed year is supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The volume has 329 explicitly typeset numbered items across chapters, repeated local numbering distinguished by PDF page.
• The statement names Leininger--Reid subgroups but provides no construction or complete separate reference. No additional citation is invented.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Registry status open and classification OPEN-TRIAGE; research summary explicitly cannot verify status.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.