A criterion for the failure of Ruelle's property

Ruelle proved that for quasiconformal deformations of cocompact Fuchsian groups, the Hausdorff dimension of the limit set is an analytic function of the deformation. In this paper, we give a criterion for the failure of analyticity for certain infinitely generated groups. In particular, we show that it fails for any infinite abelian cover of a compact surface, answering a question of Astala and Zinsmeister.