Problem 2.14 — Give a proof of Theorem 2.13 which does not depend on the classification of finite simple groups.

Definitions and mathematical statement

The principal task is to prove the source's Theorem 2.13 without dependence on the classification of finite simple groups (CFSG). The proof must retain the theorem's actual hypotheses and conclusion and avoid using CFSG-dependent results as unexamined substitutes. This does not by itself require that the proof be elementary in every other sense.

Theorem 2.13 is not included in the supplied record. Consequently its mathematical assertion cannot be restated or identified with an automorphism or normal-subgroup theorem from the speculative summary. The correct formal target remains a CFSG-independent proof of that specific, presently missing, theorem.

The continuation additionally asks to understand the frequency of genera $g$ for which the lower bound in equation (7) is attained. The bound, the quantity being bounded, the genus range and the intended frequency notion are absent. Once they are supplied, this asks for a description of the occurrence set and its frequency; existence of a natural density or another particular asymptotic is not asserted.

Short English statement

Find a proof of the missing Theorem 2.13 that avoids CFSG, and study how often the missing equation (7) lower bound is attained as the genus varies.

Sources

• B. Farb, chapter in Problems on Mapping Class Groups and Related Topics, Problem 2.14 and following sentence, supplied PDF page 28. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. The referenced Theorem 2.13 and equation (7) are absent from the supplied record.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000016$, exact source code catalogue entry. Its OPEN-TRIAGE summary speculates about the theorem's subject and reports no elementary proof through 2026, without an identifying theorem or primary citation.