Variable Critical Exponents on a Fixed Free-Deck Regular Cover

An AIM problem asks for an infinitely generated Fuchsian group of the first kind whose critical exponent is nonconstant on its quasiconformal Teichmüller space. We give an explicit affirmative construction. For a closed surface \(S_g\) , let \(K\) be the kernel of the epimorphism \(\pi_1(S_g)\to F_g\) that kills a cut system and maps its dual curves to a free basis. For every marked hyperbolic metric \(m\) , the group \(\Gamma_m=\rho_m(K)\) is infinitely generated and of the first kind. Its free deck group is nonamenable, so Brooks's theorem gives \(\delta(\Gamma_m)<1\) . When all curves of the killed cut system are pinched to length \(\ell\) , a compactly supported cutoff in one lifted cell gives \[ \lambda_0(\mathbb H^2/\Gamma_m) \leq\frac{g\sinh(1)}{\pi(g-1)}\ell, \qquad 1-\delta(\Gamma_m) \leq\frac{2g\sinh(1)}{\pi(g-1)}\ell. \] Thus the exponents tend to one while remaining strictly below one at every finite metric. All structures lie in the same quasiconformal Teichmüller space. This note is unrefereed and makes no absolute priority claim.