1. Preliminaries. Let G be a non-elementary finitely generated Fuchsian group of the second kind acting on the unit disc Δ, and let Λ(G) be the limit set of G. It is well known that Λ(G) is a perfect non-dense set on the boundary of A. The Hausdorff dimension of the limit set Λ(G) of G is defined as the non-negative number
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100063337/resource/name/S0305004100063337_eqnU1.gif?pub-status=live)
where mt(Λ(G)) denotes the t-dimensional measure of Λ(G) ([2], [3]). We say G has the type (g; m) if S = Δ/G is obtained from a compact surface of genus g by removing m(≥ 0) conformal discs.