Uniform critical-exponent control under quasiconformal deformation

Definitions and mathematical statement

Problem 4.5 is extracted as asking whether a quasiconformal deformation of $\Gamma_0$ can "bend a variation of $\delta(\Gamma_n)$ independent of $n$." The wording is corrupted, the groups $\Gamma_n$ are not defined locally, and the next heading "5. Other questions" is concatenated. The clean field repeats the malformed phrase. Thus no unique exact-independence assertion is reconstructed. The summary says it separates three plausible readings but does not list them; uniform estimates, asymptotic convergence and exact equality of variations must in any event remain distinct.

The supplied conditional result uses an increasing exhaustion $\Gamma_n$ of an infinitely generated Fuchsian group $\Gamma=\bigcup_n\Gamma_n$ by finite-type stages, with all sufficiently large stages non-elementary. For a discrete group $G$ acting on hyperbolic space, define its critical exponent by \[ \delta(G)=\inf\left\{s>0:\sum_{g\in G}e^{-s\,d(o,go)}<\infty\right\}, \] independent of the basepoint. The deformation framework must specify whether conjugates remain Fuchsian or lie in the larger Kleinian category. The aid states its distortion bound for a non-elementary Kleinian group $G$ and its $K$-quasiconformal conjugate $G'$, with $K\geq1$ and $d=\delta(G)$: \[ \frac{2d}{2K-(K-1)d}\leq\delta(G') \leq\frac{2Kd}{2+(K-1)d},\qquad K^{-1}d\leq\delta(G')\leq Kd. \] Here $0<d\leq2$ is the Kleinian ambient range. The same envelope is reported uniformly over all applicable exhaustion stages; an elementary stage with zero exponent is not included in logarithmic conclusions. The precise quasiconformal sphere-conjugacy and critical-exponent/dimension theorems needed to apply Astala distortion are not supplied.

Let $t$ range over a compact parameter space $T$, and suppose the deformations are continuous in relative quasiconformal dilatation, meaning that the dilatation needed to compare parameters tends to $1$ as they approach each other. Put $\delta_n(t)=\delta(\Gamma_n(t))$ and $\delta_\infty(t)=\delta(\Gamma(t))$. The aid reports \[ \sup_{t\in T}|\delta_n(t)-\delta_\infty(t)|\longrightarrow0. \] For a fixed base parameter $t_0$, it also reports uniform convergence of increments, \[ \sup_{t\in T}|(\delta_n(t)-\delta_n(t_0)) -(\delta_\infty(t)-\delta_\infty(t_0))|\longrightarrow0. \] These conclusions require the exhaustion, discreteness and relative-dilatation hypotheses, not just pointwise continuity of unrelated groups.

For real parameters, if the relative dilatation obeys $K(s,t)\leq e^{L|t-s|}$, the reported sharper regularity consequence is \[ |\log\delta_n(t)-\log\delta_n(s)|\leq L|t-s|, \qquad |(\log\delta_n)'(t)|\leq L\text{ almost everywhere}, \] for the positive-exponent stages. This gives a common control, not exact $n$-independence or a common nonzero derivative. No deformation with exactly equal nonzero stage variations is supplied. The status of the intended original question remains unknown in the aid; the registry's partially-solved label refers to this conditional progress.

Short English statement

The source does not define a unique meaning of variation independent of the exhaustion index. The aid supplies conditional distortion bounds and uniform convergence of stage exponents and increments, plus a uniform logarithmic Lipschitz estimate. These are not exact equality of the variations or a construction with prescribed nonzero change.

Sources

• Local UnsolvedMath registry, permanent ID 20003118, source code catalogue entry; aim-topology-notes.json, notes[205].
• AIM, Lipschitz metric on Teichmueller space, Problem 4.5, attributed in the corrupted extraction to "F. Guértitaud": https://aimath.org/pastworkshops/lipschitzteichproblems.pdf. The attribution spelling and wording require source review; no proposer identity is silently substituted.
• Astala distortion is named in the research summary; full critical-exponent, dimension and exhaustion references are not supplied.
• Machine-generated research attempt 1: conditional result, original problem status unknown at the run, rigor labelled "proved," low novelty confidence; registry partially solved. Not peer reviewed; independently verify before citation. Registry updated 14 August 2026.