Elliptic elements in Möbius groups. (English) Zbl 1180.20041
The discreteness of Möbius groups is a fundamental problem which has been discussed by many authors. In 1976, T. Jørgensen established a discreteness criterion by using the well-known Jørgensen inequality [Am. J. Math. 98, 739-749 (1976; Zbl 0336.30007)]. This criterion has become standard in the literature. Afterwards many authors tried to generalize it.
In this paper, the author, under a restriction on \(\dim(M_G)\), gives the discreteness criterion by the following theorem.
Theorem: Let \(G\subset\text{Isom}(H^n)\) be a nonelementary subgroup and \(\dim(M_G)\) be even. If \(G\) contains an elliptic element, then \(G\) is discrete if and only if \(WY(G)\) is discrete and each nonelementary subgroup of \(G\) generated by two elliptic elements is discrete.
In this paper, the author, under a restriction on \(\dim(M_G)\), gives the discreteness criterion by the following theorem.
Theorem: Let \(G\subset\text{Isom}(H^n)\) be a nonelementary subgroup and \(\dim(M_G)\) be even. If \(G\) contains an elliptic element, then \(G\) is discrete if and only if \(WY(G)\) is discrete and each nonelementary subgroup of \(G\) generated by two elliptic elements is discrete.
Reviewer: Sebahattin Ikikardes (Balıkesir)
MSC:
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
| 30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |
| 30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |
| 30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |
| 30C62 | Quasiconformal mappings in the complex plane |
Citations:
Zbl 0336.30007References:
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