zbMATH — the first resource for mathematics

The mixed elliptically fixed point property for Kleinian groups. (English) Zbl 0815.30032
A Kleinian group \(G\subset \text{PSL}(2,{\mathbf C}\) is said not to satisfy the mixed elliptically fixed point (MEFP) property if there is an elliptic element \(h\in G\) which is not contained in any degenerate subgroup of \(G\) and which satisfies the following trichotomy: either \(h\) has its fixed point set in \({\mathbf C}\) lying entirely in the limit set of \(G\) and not fixed by one loxodromic element of \(G\), or by two parabolic elements, or one of the fixed points of \(h\) lying in the limit set is not fixed by a parabolic element and the other one belongs to the discontinuity domain.
Say that \(G\) does satisfy the MEFP if the above situation does not hold.
The author proves that the MEFP-property is conserved by the Klein-Maskit combination method, as a consequence all function groups on the plane satisfy such a property. Also all geometrically finite Kleinian groups satisfy the MEFP-property. The author suggested an example of a Web group (due to B. Maskit) without this property.

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
Full Text: EMIS EuDML