Limit points via Schottky pairings. (English) Zbl 0767.30034

Nicholls, P. J.; Waterman, P. L.

Discrete groups and geometry, Proc. Conf., Birmingham/UK 1991, Lond. Math. Soc. Lect. Note Ser. 173, 190-195 (1992).

[For the entire collection see Zbl 0746.00069.]
This paper considers limit points (taken to lie at \(\infty\in\mathbb{R}^ n\cup\{\infty\}\)) for Möbius transformations acting on the upper half- space model of hyperbolic \(n\)-space. The methods used involve Schottky groups appropriately generalized to \(n\)-space. The computations are carried out in the Vahlen-Ahlfors description of hyperbolic isometries as Clifford algebra-valued Möbius transformations. Isometric spheres are used to construct infinitely generated Schottky groups for which \(\infty\) is a Dirichlet or Garnett limit point. This is then used to prove that in hyperbolic 4-space there is a parabolic fixed point which is a Garnett limit point.

Reviewer: W.Abikoff (Storrs)

MSC:

30F35

Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)

Keywords:

Dirichlet point; limit points; Möbius transformations; Schottky groups; hyperbolic isometries; parabolic fixed point; Garnett limit point