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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.

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