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.