Problem 6.2 — If G< MCG (S) is geometrically finite with cusp groups H1,...,H n, what can be said about the geometric properties of…

Definitions and mathematical statement

Let $G<\operatorname{MCG}(S)$ be geometrically finite with cusp groups $H_1,\ldots,H_n$, in the definitions intended by the source. Retain the associated extension $\Gamma_G$; in the usual surface-extension setting, \[ 1\longrightarrow\pi_1(S)\longrightarrow\Gamma_G \overset{p}{\longrightarrow}G\longrightarrow1, \qquad\Gamma_{H_i}=p^{-1}(H_i). \] The surface, extension construction, geometric-finiteness definition, and cusp hypotheses are omitted and must be recovered.

Problem 6.2. Determine the geometric properties of $\Gamma_G$. Does it have useful large-scale geometric properties, and does it satisfy some form of quasi-isometric rigidity? The source specifies neither the desired property nor a rigidity conclusion or comparison class. These remain a research program; no hyperbolicity, particular rigidity up to commensurability, or uniform theorem is silently imposed.

The source expressly warns that $\Gamma_G$ is not relatively hyperbolic with the displayed subgroups $\Gamma_{H_1},\ldots,\Gamma_{H_n}$ as peripherals, because they and their conjugates all contain the normal kernel $\pi_1(S)$. In the usual setting where this kernel is infinite and the peripherals are proper, the relevant standard obstruction is almost malnormality: \[ P_i\cap\gamma P_j\gamma^{-1}\text{ is finite unless } i=j\text{ and }\gamma\in P_i. \] All the stated preimages contain the common infinite kernel, preventing that condition in the nondegenerate peripheral setup. The source's phrase "any two conjugates" needs this exception for the same peripheral and its internal conjugation. The excerpt does not address degenerate peripheral choices or finite-kernel cases. This warning excludes the specified peripheral family, not every conceivable relative-hyperbolic description.

Short English statement

Investigate large-scale geometry and possible quasi-isometric rigidity of extensions associated to geometrically finite mapping-class subgroups. The source rules out the obvious cusp-preimage peripheral structure because the preimages share the infinite surface-group kernel.

Sources

• Local UnsolvedMath registry, permanent ID 11000242, source code catalogue entry, Problem 6.2, PDF page 278. No individual proposer is supplied.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The volume contains 329 explicitly typeset headings across contributed chapters, with repeated labels disambiguated by page.
• The shared-kernel relative-hyperbolicity obstruction is part of the supplied statement, without a separate cited theorem or earlier extension definition.
• Supplied status open and OPEN-TRIAGE. The research summary says only "Truncated." Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.