# Discrete groups and thin sets

**Ann. Acad. Sci. Fenn. A I**Vol. 23 (1998), p. 291-315.

[Artikel, refereegranskad vetenskaplig]

Let $\Gamma$ be a discrete group of M\"obius transformations acting on and preserving the unit ball in $\Rdim$ (i.e.\ Fuchsian groups in the planar case). We will put a hyperbolic ball around each orbit point of the origin and refer to their union as the {\em archipelago of $\Gamma$}. The main topic of this paper is the question: ``How big is the archipelago of $\Gamma$?'' We will study different ways to answer various meanings of that question using concepts from potential theory such as {\em minimal thinness} and {\em rarefiedness} in order to give connections between the theory of discrete groups and small sets in potential theory. One of the answers that will be given says that the critical exponent of $\Gamma$ equals the Hausdorff dimension of the set on the unit sphere where the archipelago of $\Gamma$ is not minimally thin. Another answer tells us that the limit set of a geometrically finite Fuchsian group $\Gamma$ is the set on the boundary where the archipelago of $\Gamma$ is not rarefied.

**Nyckelord: **Discrete group, Fuchsian group,
Kleinian group, Poincar\'e series, horocycle, limit set,
minimal thinness, reduced function, rarefiedness

*The Mathematical Review of this paper:
http://www.ams.org/mathscinet/pdf/1642114.pdf?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=9&s4=Lundh&s5=&s6=&s7=&s8=All&yearRangeFirst=&yearRangeSecond=&yrop=eq*

Denna post skapades 2006-12-05. Senast ändrad 2014-09-02.

CPL Pubid: 23855