Pinching a Free Normal Cover: Variable Critical Exponent in a Quasiconformal Teichmuller Space

Summary

Take a closed hyperbolic surface and cut it along enough disjoint curves to leave a connected sphere with holes. A fixed normal cover then arranges copies of this cut-open surface like the vertices of a nonabelian free group.

This anonymous, unrefereed candidate argues that the cover is always uniformised by an infinitely generated Fuchsian group of the first kind. Its critical exponent is below $1$ for every marked compact-base metric. When all cut curves are pinched to a common small length $\ell$, the exponent approaches $1$ from below with

It therefore assumes infinitely many values inside one all-Fuchsian quasiconformal deformation family.

The historical boundary matters. Astala and Zinsmeister (1995) appear already to answer the existential AIM question under a broad quasi-Fuchsian group-deformation reading. The present claim is an all-Fuchsian strengthening and a direct solution candidate only under the reduced surface-based reading. It is not presented as the first solution.

Why the problem matters

The critical exponent measures exponential orbit growth. For finitely generated Fuchsian groups it is closely tied to the geometry of the limit set, but infinitely generated first-kind groups can have the entire boundary circle as limit set while still having exponent below $1$.

AIM Problem 4.3 asks whether the exponent can vary within one reduced quasiconformal Teichmuller space. An all-Fuchsian answer is especially sharp: the variation occurs while every group continues to preserve a hyperbolic plane, rather than by moving into three-dimensional quasi-Fuchsian geometry.

The exact theorem candidate

For every integer $g\ge 2$, choose a geometric symplectic basis $(a_i,b_i)$ of a closed genus-$g$ surface so that the $b_i$ form a disjoint cut system. Define

For a marked closed hyperbolic metric $m$, let $\rho_m$ be its Fuchsian holonomy and set $\Gamma_m=\rho_m(K)$. The candidate theorem states:

• every $\Gamma_m$ is infinitely generated and of the first kind;
• the marked covers give points in one reduced surface-based space $T_{qc}^{\mathrm{red}}(X_0)$ and, by reflected sphere extensions, in the all-Fuchsian locus of the AIM group space $T_{qc}(\Gamma_0)$;
• $\delta(\Gamma_m)<1$ for every marked compact-base metric;
• along simultaneous pinching of all $b_i$ to length $\ell$, for sufficiently small $\ell$,

No monotonicity, exact exponent formula, global identification of deformation spaces, injective parametrisation, or arbitrary quasi-Fuchsian theorem is claimed. The value $\ell=0$ is a boundary degeneration, not a point of the family.

How the construction works

The epimorphism $q$ makes the cover regular with deck group $F_g$. A nontrivial normal subgroup of the cocompact surface lattice has the full boundary circle as limit set, so $\Gamma_m$ is of the first kind. If it were finitely generated, absence of parabolics would make it convex cocompact; a full limit set would then force a compact infinite-sheeted cover, a contradiction.

Marked quasiconformal maps between compact base surfaces lift equivariantly to the fixed kernel. Reflection extends the lifted disk maps to sphere maps, which places the resulting group points in the all-Fuchsian locus of the AIM space. Marked Möbius equivalence preserves the critical exponent.

Dougall and Sharp's normal-subgroup theorem applies because the ambient surface group is convex cocompact, the subgroup is normal, and the quotient $F_g$ is nonamenable. It gives the strict inequality $\delta(\Gamma_m)<1$ at every compact-base metric.

The fixed-width pinching calculation

After cutting along the $b_i$, copies of one compact cell are indexed by $F_g$. A collar about a cut curve of length $\ell$ has half-width tending to infinity as $\ell\to0$. On the $2g$ half-collars adjacent to one selected cell, use a function that rises linearly from $0$ to $1$ across width exactly $1$ and then remains $1$ on the rest of that cell.

The total Dirichlet energy is

while the plateau area is at least

For small $\ell$, the denominator stays bounded below and the Rayleigh quotient is $O_g(\ell)$. Sullivan's bottom-of-spectrum formula then converts this into $1-\delta=O_g(\ell)$. Combined with the strict inequality below one, convergence rules out a finite set of exponent values.

What was checked and replayed

The release package performs deterministic producer-side checks. It does not claim that finite computation proves the universal theorem.

• Forty-two collar-model samples check the exact energy, plateau, Rayleigh and spectral-conversion formulas.
• Sixteen unit and hostile-mutation tests reject wrong collar counts, nonpositive lengths, genus one, amenable deck rank, extra collars and broken seam conditions, in normal and optimized Python modes.
• Source-parity checks bind theorem-critical statements across LaTeX, Markdown and the claim ledger.
• PDF gates check required reader text, raw-TeX leakage, qpdf structure and a complete normalized text digest.
• A 55-file manifest and an outer checksum bind the complete public archive; a clean extraction passes the full read-only replay.
• Public GitHub Actions rebuild the reader PDF from source before repeating the normal and optimized replay.

Evidence and assurance boundary

The written manuscript is the evidence for the general mathematical claims. The finite program checks explicit identities, implementation invariants and representative samples; it does not replace the proof or verify the imported theorems.

The supplied review was actioned point by point. Five producer-coordinated editorial roles initially held the package for repair. After the deformation- space bridge, closest-prior-work positioning, exact theorem-hypothesis maps, portable build, accessibility language and release receipts were repaired, one exact-archive domain confirmation returned PASS_WITH_NOTES at reported confidence $0.94$.

Those are internal editorial records, not authenticated unaffiliated specialist review. Public availability, deterministic replay, independent rerun, independent reimplementation, formal verification, specialist review, editorial peer review, novelty and historical priority remain distinct.

Limitations and what remains open

• Under a broad group-deformation reading, the existential AIM question appears already affirmative through Astala and Zinsmeister's quasi-Fuchsian family.
• The exact all-Fuchsian construction may be specialist folklore or an unstated consequence of known results; the bounded search cannot establish novelty or priority.
• No monotonicity or exact formula for $\delta(\Gamma_\ell)$ is proved.
• The compact-base family is not claimed to embed injectively into the infinite-type Teichmuller space.
• The reflected extension supplies all-Fuchsian points; it does not prove a theorem for arbitrary quasi-Fuchsian deformations.
• No unaffiliated reconstruction, proof-assistant formalisation, authenticated external specialist review or journal peer review is attached.
• The PDF is untagged. This structured web page and paper.md provide text alternatives, but mathematical notation is linearised rather than encoded with full semantic accessibility.

Who should care

The release is aimed at researchers in Fuchsian and Kleinian groups, hyperbolic surfaces, regular covers, spectral geometry and infinite-type Teichmuller theory. It may also be useful to reviewers studying how a short bridge among classical theorems should expose its interpretation, dependency and priority risks before candidate publication.

Where to inspect and replay

Start with the PDF for the complete proof. In the archive, CLAIM_SCOPE.md states the exact theorem and exclusions, SOURCES.md maps every imported theorem to its use, NOVELTY_REPORT.md records the broad-reading precedent, and reviews/internal/ preserves the role-separated editorial trail. Run bash run_all.sh from the repository root for the finite replay and package gates.

The GitHub Actions run linked in the assurance panel is the public clean- checkout reconstruction. MANIFEST.sha256 inventories the 55 non-circular payload files, while release-level SHA256SUMS binds the downloadable ZIP and PDF.

Next work

The highest-value next step is an unaffiliated mathematical reconstruction of the reflected group-space bridge and the fixed-width Rayleigh argument. A Fuchsian-group specialist should then assess the precise AIM interpretation and closest prior art. Further mathematical work could seek monotonicity or a sharper asymptotic for the exponent, while a formalisation should keep the elementary collar calculation separate from the imported normal-subgroup and spectrum theorems.

Paper, archive, and package map

• Paper: the canonical 9-page PDF contains the complete argument, hypothesis maps, positioning and bibliography.
• Archive: the ZIP contains source, accessible text, structured claims, reviews, verification programs, receipts, licences and the complete manifest.
• Repository: the annotated candidate tag fixes the reviewed source and hosted replay workflow.
• Zenodo: the version DOI archives the same ZIP, PDF and checksum sidecar.
• Licensing: original prose and data are CC0-1.0; original code, tests and workflows are MIT; third-party sources and the bundled OFL font are credited but not relicensed.

@misc{pinchingfreenormalcover2026,
  title        = {Pinching a Free Normal Cover: Variable Critical Exponent in a Quasiconformal Teichmuller Space},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22229561},
  url          = {https://doi.org/10.5281/zenodo.22229561},
  version      = {0.2.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/pinching-free-normal-cover/}
}