The exponent of convergence of Poincaré series associated with some discontinuous groups (Q1113323)

The following theorem is proved. Suppose \(\Gamma\) is a discontinuous group of conformal motions of the \((n+1)\)-dimensional ball \(B^{n+1}\) and assume that the limit set \(\Lambda\) (\(\Gamma)\) has more than two points. Further assume \(\xi_ 0\in \Lambda (\Gamma)\) is a parabolic fixed point, that is, \(\xi_ 0\) is the only fixed point of some \(\gamma\in \Gamma\) and that the isotropy group of \(\xi_ 0\) has a rank k free abelian subgroup. Then the exponent of convergence of \(\Gamma\) is at least k and this estimate is best possible.