Quasiconformal groups with small dilatation. II.

*(English)*Zbl 1246.30074The authors investigate to what extent quasiconformal groups are similar or not to Möbius groups.

Denote by \(\overline{\mathbb B}^n\) the unit closed ball in the real \(n\)-dimensional Euclidean space and by \(\mathbb S^{n-1}\) its boundary. In the case \(n=2\), every quasiconformal group acting on \(\overline{\mathbb B}^n\) is the quasiconformal conjugate of a conformal group or a Möbius group, as proved independently in [D. Sullivan, Ann. Math. Stud. 97, 465–496 (1981; Zbl 0567.58015)] and [P. Tukia, Ann. Acad. Sci. Fenn., Ser. A I Math. 5, No. 1, 73–78 (1980; Zbl 0411.30038)]. On the other hand, for \(n\geq 3\) there exists a discrete quasiconformal group not quasiconformally conjugate to a Möbius group. This was first proved in [P. Tukia, Ann. Acad. Sci. Fenn. Ser. A I, Math. 6, No. 1, 149–160 (1981; Zbl 0443.30026)] and refined later in [the second author, Ann. Acad. Sci. Fenn. Ser. A I, Math. 11, No. 2, 179–202 (1986; Zbl 0635.30021)].

The basis for a deeper study of quasiconformal groups in case \(n\geq 3\) was established in [F. W. Gehring and the second author, Proc. Lond. Math. Soc. 55, 331–358 (1987; Zbl 0628.30027)]. More recently, the authors published the immediate precedent of the article under review [Proc. Am. Math. Soc. 129, No. 7, 2019–2029 (2001; Zbl 0984.30023)].

For a fixed real number \(K\geq 1\) it is said that a discrete group \(G\) of orientation-preserving homeomorphisms of \(\overline{\mathbb B}^n\) or \(\mathbb S^{n-1}\) is a \(K\)-quasiconformal group if each of its elements is \(K\)-quasiconformal. The group \(G\) is said to be quasiconformal if it is \(K\)-quasiconformal for some \(K\). The infimum of such values \(K\) is the dilatation of \(G\), and \(G\) is a Möbius group if its dilatation is 1. The authors introduce the notion of \((M, \delta)\)-presented quasiconformal group, where \(M\) is a positive integer and \(\delta >0\) is a real number, as those satisfying the following conditions:

1) \(G\) is generated by \(m\leq M\) of its elements, say \(S=\{g_1, \dots, g_m\}\).

2) Every simple relation in the generators of \(S\) has word length at most \(M\).

3) The order of every element of \(G\) of finite order is at most \(M\).

4) The hyperbolic distance between the origin \(0\) and their images \(g_i(0)\), for \(1\leq i\leq m\), is smaller that \(\delta^{-1}\).

Of course, this last condition must be modified in dealing with homeomorphisms of \(\mathbb S^{n-1}\).

One of the main results of the paper under review states that each nonelementary \((M, \delta)\)-presented quasiconformal group with small enough dilatation (i.e. there exists a bound for its dilatation depending only on the number of generators and the length of the relations providing a presentation of it) is isomorphic to a Möbius group. As usual, the group \(G\) is said to be nonelementary if its limit set \(\Lambda (G)\) contains more than two points.

One can ask if the isomorphism between the groups above is induced by a homeomorphism between the limit sets. The authors answer this question in the affirmative, under not so strong assumptions, in Theorem 5.4. They have also included in this carefully written article some enlightening examples and challenging open questions.

Denote by \(\overline{\mathbb B}^n\) the unit closed ball in the real \(n\)-dimensional Euclidean space and by \(\mathbb S^{n-1}\) its boundary. In the case \(n=2\), every quasiconformal group acting on \(\overline{\mathbb B}^n\) is the quasiconformal conjugate of a conformal group or a Möbius group, as proved independently in [D. Sullivan, Ann. Math. Stud. 97, 465–496 (1981; Zbl 0567.58015)] and [P. Tukia, Ann. Acad. Sci. Fenn., Ser. A I Math. 5, No. 1, 73–78 (1980; Zbl 0411.30038)]. On the other hand, for \(n\geq 3\) there exists a discrete quasiconformal group not quasiconformally conjugate to a Möbius group. This was first proved in [P. Tukia, Ann. Acad. Sci. Fenn. Ser. A I, Math. 6, No. 1, 149–160 (1981; Zbl 0443.30026)] and refined later in [the second author, Ann. Acad. Sci. Fenn. Ser. A I, Math. 11, No. 2, 179–202 (1986; Zbl 0635.30021)].

The basis for a deeper study of quasiconformal groups in case \(n\geq 3\) was established in [F. W. Gehring and the second author, Proc. Lond. Math. Soc. 55, 331–358 (1987; Zbl 0628.30027)]. More recently, the authors published the immediate precedent of the article under review [Proc. Am. Math. Soc. 129, No. 7, 2019–2029 (2001; Zbl 0984.30023)].

For a fixed real number \(K\geq 1\) it is said that a discrete group \(G\) of orientation-preserving homeomorphisms of \(\overline{\mathbb B}^n\) or \(\mathbb S^{n-1}\) is a \(K\)-quasiconformal group if each of its elements is \(K\)-quasiconformal. The group \(G\) is said to be quasiconformal if it is \(K\)-quasiconformal for some \(K\). The infimum of such values \(K\) is the dilatation of \(G\), and \(G\) is a Möbius group if its dilatation is 1. The authors introduce the notion of \((M, \delta)\)-presented quasiconformal group, where \(M\) is a positive integer and \(\delta >0\) is a real number, as those satisfying the following conditions:

1) \(G\) is generated by \(m\leq M\) of its elements, say \(S=\{g_1, \dots, g_m\}\).

2) Every simple relation in the generators of \(S\) has word length at most \(M\).

3) The order of every element of \(G\) of finite order is at most \(M\).

4) The hyperbolic distance between the origin \(0\) and their images \(g_i(0)\), for \(1\leq i\leq m\), is smaller that \(\delta^{-1}\).

Of course, this last condition must be modified in dealing with homeomorphisms of \(\mathbb S^{n-1}\).

One of the main results of the paper under review states that each nonelementary \((M, \delta)\)-presented quasiconformal group with small enough dilatation (i.e. there exists a bound for its dilatation depending only on the number of generators and the length of the relations providing a presentation of it) is isomorphic to a Möbius group. As usual, the group \(G\) is said to be nonelementary if its limit set \(\Lambda (G)\) contains more than two points.

One can ask if the isomorphism between the groups above is induced by a homeomorphism between the limit sets. The authors answer this question in the affirmative, under not so strong assumptions, in Theorem 5.4. They have also included in this carefully written article some enlightening examples and challenging open questions.

Reviewer: Jose Manuel Gamboa (Madrid)

##### MSC:

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

57S30 | Discontinuous groups of transformations |

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\textit{P. Bonfert-Taylor} and \textit{G. Martin}, Complex Var. Elliptic Equ. 51, No. 2, 165--179 (2006; Zbl 1246.30074)

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