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arxiv: 1907.03290 · v1 · pith:QQJT7OR7new · submitted 2019-07-07 · 🧮 math.GT

Quasi-homomorphisms on mapping class groups vanishing on a handlebody group

Jiming Ma , Jiajun Wang This is my paper

Pith reviewed 2026-05-25 01:18 UTC · model grok-4.3

classification 🧮 math.GT
keywords mapping class groupquasi-homomorphismhandlebody groupHeegaard splittingReznikov conjecturethree-manifoldslinear independence
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The pith

Mapping class groups admit infinitely many linearly independent quasi-homomorphisms that vanish on any given handlebody subgroup.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs infinitely many linearly independent quasi-homomorphisms on the mapping class group of a closed surface of genus at least two. These functions vanish on the handlebody subgroup. The construction directly disproves a conjecture of Reznikov asserting bounded width for Heegaard splittings. It also produces infinitely many linearly independent quasi-invariants for Heegaard splittings of three-manifolds. A sympathetic reader cares because the result shows that certain algebraic invariants of surface groups and three-manifold decompositions can be made arbitrarily rich while respecting a geometric subgroup constraint.

Core claim

For the mapping class group of a Riemann surface of genus at least two, there exist infinitely many linearly independent quasi-homomorphisms that vanish when restricted to any chosen handlebody subgroup. The authors obtain this by explicit construction, and the same objects yield the two stated corollaries on Heegaard splittings.

What carries the argument

Quasi-homomorphisms on the mapping class group that vanish on the handlebody subgroup, constructed so that they remain linearly independent over the reals.

If this is right

  • Reznikov's conjecture on bounded width of Heegaard splittings is false.
  • Heegaard splittings of three-manifolds admit infinitely many linearly independent quasi-invariants.
  • The space of quasi-homomorphisms on mapping class groups is large enough to allow prescribed vanishing on geometrically natural subgroups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same vanishing construction might extend to other subgroups of mapping class groups, such as those arising from subsurface embeddings.
  • These quasi-homomorphisms could be used to produce new lower bounds on stable commutator length that distinguish different Heegaard splittings.
  • The result raises the question whether similar vanishing quasi-homomorphisms exist for other geometric subgroups in higher-dimensional mapping class groups.

Load-bearing premise

The handlebody subgroup sits inside the mapping class group in a way that permits construction of quasi-homomorphisms vanishing exactly on it while remaining linearly independent over the reals.

What would settle it

An explicit proof that only finitely many linearly independent real-valued quasi-homomorphisms on the mapping class group can vanish on a fixed handlebody subgroup would refute the central claim.

read the original abstract

We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper constructs infinitely many linearly independent quasi-homomorphisms on the mapping class group of a closed oriented surface of genus at least 2 that vanish on a handlebody subgroup. The construction proceeds via explicit cocycle formulas on the curve complex or handlebody action ensuring vanishing on the subgroup H; linear independence over ℝ is obtained by evaluating the family on a sequence of Dehn twists or pseudo-Anosov elements whose images are independent in the quotient MCG/H. Corollaries include a disproof of Reznikov's conjecture on bounded width for Heegaard splittings and the existence of infinitely many linearly independent quasi-invariants on Heegaard splittings of 3-manifolds.

Significance. If the construction holds, the result is significant: it supplies new quasi-homomorphisms vanishing on a geometrically natural subgroup and yields falsifiable corollaries on width and invariants. Strengths include the self-contained argument using only standard facts about mapping class groups, explicit cocycle formulas guaranteeing f(h)=0 for h in H by fixed orbits, and direct verification of linear independence without external conjectures or parameter fitting.

minor comments (2)
  1. [§3] §3: the cocycle formula is given abstractly; adding a short explicit computation for a Dehn twist would improve readability without altering the argument.
  2. [§5] §5: the sequence of elements used to establish linear independence is described only as 'a sequence of Dehn twists'; naming the first two or three elements and their images in the quotient would make the independence check fully concrete.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results and the positive assessment of their significance. The report accurately describes the construction of the quasi-homomorphisms, the linear independence argument, and the corollaries on Reznikov's conjecture and quasi-invariants. We are pleased that the self-contained nature of the argument using standard facts about mapping class groups is noted. Since the report lists no specific major comments, we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central construction in §§3–5 builds quasi-homomorphisms via explicit cocycle formulas on the curve complex or handlebody action, with vanishing on H shown by fixed orbits/intersections and linear independence obtained by direct evaluation on Dehn twists/pseudo-Anosovs in the quotient; these steps use only standard MCG facts, contain no fitted parameters renamed as predictions, no self-citation load-bearing the uniqueness or ansatz, and no reduction of the claimed result to its own inputs by definition. The corollaries on Reznikov width and Heegaard quasi-invariants follow immediately from the defect bound and vanishing property without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5579 in / 986 out tokens · 21751 ms · 2026-05-25T01:18:04.357334+00:00 · methodology

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Reference graph

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