# Verification report — AMR-109-0020 (Farb, Problem 2.19: the example question on nilpotent automorphism groups)

Verification date: 2026-10-14 (UTC; the runs were made on 13 and 14 October).

**Verdict.** The note answers the **example question** of Problem 2.19 of Farb's list with **no**: the surface of
genus g whose automorphism group is nilpotent of the largest possible order 16(g−1) is not unique.
- **Theorem 1.1.** There are at least four pairwise non-isomorphic compact Riemann surfaces of genus 9 whose full
  automorphism group is a 2-group of order 128 = 16(g−1). They are H/K_i for the kernels K_i of four smooth
  epimorphisms of the triangle group L = Δ(2,4,8) onto groups G_1, …, G_4 of order 128 given by presentations from
  Conder's list of large group actions.
- **The full automorphism group.** Lemma 2.1 proves that Δ(2,3,8) is the only Fuchsian group properly containing
  Δ(2,4,8) (index 3); explicit certificates show that no K_i is normal in it, so Aut(H/K_i) = L/K_i.
- **Theorem 1.2 (computer-assisted).** Under the completeness of Conder's list, the number of surfaces of genus g
  whose full automorphism group is a 2-group of order 16(g−1) is 0, 1, 1, 4, 3, 7, 15 for g = 2, 3, 5, 9, 17, 33,
  65. Two independent enumerations of all normal subgroups of Δ(2,4,8) with a 2-group quotient of order at most
  1024 (made in two verification runs; programs and outputs in the package) found exactly the kernels of the list
  and the mirror images of the two chiral ones, so for these genera the counts do not depend on the hypothesis.
- **Remark 4.2 (genus 5).** The genus-5 example of Zomorrodian (1987), the group L/γ₄(L) of order 64 and class 3,
  is the action whose kernel is normal in Δ(2,3,8); its surface has 192 automorphisms. The genus-5 surface with
  nilpotent full automorphism group of order 64 comes from the other action (class 4).
- **What is used from the literature.** Zomorrodian's bound 16(g−1) for nilpotent groups and its equality case (Trans. Amer. Math. Soc. 288 (1985), Theorem 1.8.4; for 2-groups also Glasgow Math. J. 29 (1987), Theorem 1.1.2) is used only to identify 16(g−1) as the largest order; the four surfaces do not depend on it. The presentations are Conder's; everything
  used about them is re-derived from the printed relators.
- **Not new.** The question is Farb's; the bound, the equality condition and the existence of 2-group actions are
  Zomorrodian's; the four groups of order 128 (SmallGroup(128,75), (128,134), (128,136), (128,138)) are listed with
  their generating vectors in the LMFDB, so the non-uniqueness can be read off from public data once the groups are
  known to be full automorphism groups and pairwise non-isomorphic. The inclusion Δ(2,4,8) < Δ(2,3,8) is one of
  the inclusions of Singerman's classification.
- **New, to our knowledge and relative to the literature found.** The negative answer to the example question,
  the proof of the overgroup lemma used, the certificates, and the counts for g ≤ 65.
- **Limits.** The first, open-ended sentence of Problem 2.19 is not addressed. Genera g = 2^n + 1 with n ≥ 7 are not
  treated. Singerman's 1972 paper was not accessible; the inclusion used from it is proved in the note. The genus-2 statement depends on the reading of the
  question (Section 1.3 of the note).

## Statement check
- Problem 2.19 of Farb (arXiv:math/0606432v1, p. 16) is quoted verbatim in Section 1.1; the record AMR-109-0020
  continues with the sentences on Hurwitz surfaces that lead to the next problem of the source, which are not part
  of Problem 2.19 (the note says so).
- Reading: the example question asks whether, for g with g−1 a power of 2, exactly one surface has a nilpotent full
  automorphism group of order 16(g−1). The note discusses two other readings (a nilpotent group of automorphisms of
  that order, not necessarily the full group; the largest order of a nilpotent full automorphism group). The
  genus-9 answer is "not unique" under all three readings.

## Verification runs (three independent AI-assisted runs; none saw the others' reports)
- **Run A (statement and proofs).** Compared the reading of the question with Farb's text; re-derived Lemma 2.1
  (the hyperbolic trigonometry of (a), the Schreier generators of (b), the area enumeration and the
  elliptic-element argument of (c)), Lemmas 2.2–2.4, the logic of Proposition 3.1 and the deduction of
  Theorem 1.2; recomputed the certificates with programs of its own. It found one gap: in Lemma 2.1(c) the element
  u had to be chosen so that it moves the coset L (closed by replacing u by a conjugate). It asked for references
  for two standard facts, for a statement of what rests on the coset enumeration (the upper bound |G_i| ≤ 128) and
  what on permutations, and for wording on the conditional counts. All applied.
- **Run B (own computation and literature).** Own programs from the statements of the note (SymPy coset
  enumeration, permutation certificates, normality in Δ(2,3,8) by the order of the extension quotient and by
  Schreier rewriting, reflexibility, invariants, an exhaustive isomorphism test): every number of Tables 1 and 2
  reproduced for all 35 actions with g ≤ 65. It matched the 32 generating vectors of the LMFDB for genus 9 with the
  four kernels and identified the groups as SmallGroup(128,136), (128,134), (128,75), (128,138); it showed that the
  four groups are pairwise non-isomorphic; it enumerated all normal subgroups of Δ(2,4,8) with 2-group quotient of
  order ≤ 1024 and found exactly the census kernels (and the two mirror images in genus 65). It checked the zbMATH
  reviews of Zomorrodian's papers, the bibliography (Crossref) and the census header, and searched the literature
  (arXiv, zbMATH Open, Crossref, OpenAlex, LMFDB): no statement on the uniqueness. It required the citation of the
  LMFDB data as prior public data and an exact quotation of the review Zbl 0565.20029 (with its misprint
  g = 2^k − 1). All applied.
- **Run 2 (the third run, on the final text).** Re-derived Lemmas 2.1–2.4, Proposition 3.1 and the deduction of
  Theorem 1.2; compared every statement about the sources with Farb's text, the zbMATH reviews, Conder's file, the
  LMFDB records and Crossref. Own programs: rebuilt the four groups (order 128, generators of orders 2, 4, 8),
  kernels pairwise different also up to reflection, none normal in Δ(2,3,8); invariants recomputed; the four
  groups pairwise non-isomorphic by canonical Cayley tables over all generating pairs of orders 4 and 8; identified
  with the LMFDB groups; for all genera of Table 1 the orders, reflexibility and normality recomputed; its own
  enumeration of the 2-group quotients of Δ(2,4,8) up to order 1024 again found exactly the census kernels. It
  re-ran certificates.py and noniso.py from a copy: recorded outputs reproduced. Required: a clause of Lemma 2.3
  (L/K is a 2-group only when g−1 is a power of 2; genus 11 is a counterexample to the old wording, which was not
  used in a false way), the removal of a sentence on the passport flag of the LMFDB, the support of the sentence on
  the completeness check; recommended: a reference for the classical genus-2 list, a wording in Section 2.1, small
  clarifications in Lemma 2.1(c). All applied.

## Changes after the runs by the author
- Zomorrodian, Glasgow Math. J. 29 (1987) 237–244, was read in full in the publisher's open copy (14 October 2026):
  Theorem 1.1.2 (bound and equality case for 2-groups), Corollary 1.4.1 (class at least 3), Theorem 1.4.1
  (|L/γ₄(L)| = 64), Theorem 1.2.1 (existence) and the remark on the genus-5 example. The note now quotes the paper
  directly (Sections 1.1 and 1.4) and identifies the genus-5 example (Remark 4.2, checked with the package outputs
  and a separate computation of the nilpotency classes of the two genus-5 groups: 4 and 3).
- Zomorrodian, Trans. Amer. Math. Soc. 288 (1985) 241–255, was read in full in the publisher's free archive copy
  (14 October 2026): Theorem 1.8.4 (bound and equality case), Corollary 2.1.1, Theorem 2.1.2 (existence for every
  n ≥ 4) and the genus-2 example on p. 253. The paper speaks of "at least one surface" and says nothing about the
  number of surfaces. Sections 1.1, 1.4 and Remark 4.1 now quote it directly; the bibliography note was updated.

## Readings
- Read: Farb, arXiv:math/0606432v1, pp. 15–16, and the bibliography entries for Kulkarni and Zomorrodian; Zomorrodian 1985 in full (publisher's free archive copy, 14 October 2026: Theorem 1.8.4, Corollary 2.1.1, Theorem 2.1.2 and the genus-2 example on p. 253; no statement on the number of surfaces) and its review Zbl 0565.20029; Zomorrodian 1987 in full; the header and the entries of genus 2, 3, 5,
  9, 17, 33, 65 of Conder's file BigSurfaceActions-Genus2to101-ByGenus.txt (version of April 2026; SHA-256 in the
  note); the LMFDB records for genus 9 and group order 128.
- Not read: Singerman, J. London Math. Soc. (2) 6 (1972)
  (not accessible; the inclusion used is proved in the note); Conder's survey (Contemp. Math. 629, 2014); Katok's
  book (standard facts); Bolza 1887 (classical genus-2 list).

## Reproducibility
- `note/`: the programs and data of the note with recorded outputs; `run_note.sh` re-runs them and compares
  (RERUN_LOG.txt: all outputs reproduced).
- `verification_run_B/`: programs and outputs of run B (own checks, LMFDB match, the enumeration of 2-group
  quotients up to order 1024).
- `independent_run_2/`: programs and outputs of run 2 (groups, census, LMFDB match, its own enumeration).

This is AI-assisted, self-audited and unrefereed work, not independent human peer review or formal certification.