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Divergence groups have the Bowen property. (English) Zbl 0999.37030

A Kleinian group is a discrete group of isometries of hyperbolic three-space \({\mathbb{H}}^3\). It is called Fuchsian if it leaves invariant the unit disc \({\mathbb{D}}\subset \partial {\mathbb{H}}^3\). A deformation of a Fuchsian group is a conformal map \(\Phi:{\mathbb{D}}\to \Omega\) such that for every \(g\in G\), the map \(\Phi\circ g\circ \Phi^{-1}\) is a Möbius map, and the deformation is quasiconformal if \(\Phi\) extends to a quasiconformal map on \({\mathbb{C}}\). Now a Fuchsian group \(G\) is said to have the Bowen property if, for every quasiconformal deformation of \(G\), the limit set \(\Lambda\) either is the circle, or has dimension \(>1\). It was shown by R. Bowen [Publ. Inst. Hautes Étud. Sci. 50, 11-25 (1979; Zbl 0439.30032)] that cocompact groups, i.e. groups with \({\mathbb{D}}/G\) compact, had this property, a result extended by D. Sullivan [Bull. Am. Math. Soc. 6, 57-73 (1982; Zbl 0489.58027)] to cofinite groups, i.e. groups for which \({\mathbb{D}}/G\) has finite hyperbolic area. On the other hand, K. Astala and M. Zinsmeister [C. R. Acad. Sci., Paris, Sér. I Math. 311, 301-306 (1990; Zbl 0709.30039)] proved that convergence groups of the first kind, i.e. groups for which the series \[ \sum_{g\in G}\exp(-\rho(0,g(0)) \] converges and the limit set is the circle, do not have the Bowen property. The only remaining case of interest is that of divergence groups, and this is solved in this paper, where the author proves the following:
Theorem. Suppose \(G\) is a divergence-type Fuchsian group and \(G'=\Phi\circ G\circ \Phi^{-1}\) is a deformation. Then either \(\partial \Omega\) is a circle or \(\delta(G')>1\), where \[ \delta(G')=\inf\biggl\{s:\sum_{g\in G'}\exp(-s\rho(0,g(0)))<\infty\biggr\}.\tag{1} \] Since it is known that \(\delta(G')\leq \dim(\Lambda)\), this implies \(G\) has the Bowen property.
The proof makes use of the following result of D. Sullivan [Semin. Bourbaki, 32e annee, Vol. 1979/80, Exp. 554, Lect. Notes Math. 842, 196-214 (1981; Zbl 0459.57006)]: there is a universal constant \(K\) such that for any hyperbolic, simply connected planar domain \(\Omega\), there is a \(K\)-quasiconformal map from \(\Omega\) to the boundary of the convex hull of \(\Omega^c\) in \({\mathbb{H}}^3\). This implies that a conformal map \(\Phi:{\mathbb{D}}\to \Omega\) can be factored as \(\Phi=h\circ \Psi\) where \(\Psi \) is a \(K\)-quasiconformal self-map of \({\mathbb{D}}\) and \(h:{\mathbb{D}}\to \Omega\) is expanding, and this map \(\Psi\) is used here to estimate the series in (1).

MSC:

37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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