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Quasirigidity of hyperbolic 3-manifolds and scattering theory. (English) Zbl 0915.30037

Consider two convex cocompact torsion-free Kleinian groups \(\Gamma_1\) and \(\Gamma_2\) with nonempty domains of discontinuity \(\Omega(\Gamma_i)\) and limit sets \(L_{\Gamma_i}\), \(i=1,2\). Assume that there exists an orientation-preserving diffeomorphism \(\psi:\Omega(\Gamma_1)\to\Omega(\Gamma_2)\) which induces an isomorphism of \(\Gamma_1\) and \(\Gamma_2\). For a complex parameter \(s\), let \({\mathcal F}_s(\Gamma)\) be the space of automorphic forms of weight \(s\) on \(\Omega(\Gamma)\). For \(\text{Re }s=1\), \(\mathcal{F}_s(\Gamma)\) possesses a natural \(L^2\) inner product, so we can complete these spaces to form Hilbert spaces \(\mathcal{H}_s(\Gamma)\), where \(s=1+i\sigma\). Let \(S_i(s)\) be the scattering operator mapping \(\mathcal{F}_{2-s}(\Gamma_i)\to\mathcal{F}_s(\Gamma_i)\) for \(i=1,2\). We can use \(\psi^*\) to define a pullback of the scattering operator \(S_2(s)\) to an operator \(\psi^*S_2(s):{\mathcal F}_{2-s}(\Gamma_1)\to{\mathcal F}_s(\Gamma_1)\). Thus we can define the relative scattering operator \(S_{\text{rel}}(s)=S_1(s)-\psi^*S_2(s)\), which we regard by extension as an operator \(\mathcal{H}_{-\sigma}(\Gamma_1)\to\mathcal{H}_\sigma(\Gamma_1)\), for \(s=1+i\sigma\). The main result of the paper is the following theorem. Suppose \(\Gamma_i\), \(i=1,2\) are convex cocompact torsion-free Kleinian groups so that \(M(\Gamma_i)=\Omega(\Gamma_i)/{\Gamma_i}\) has infinite hyperbolic volume. Let \(\psi:\Omega(\Gamma_1)\to\Omega(\Gamma_2)\) be an orientation-preserving \(C^\infty\)-diffeomorphism conjugating \(\Gamma_1\) to \(\Gamma_2\). Fix \(s\in {\mathbb{C}}:Re s=1\) \(s\neq 1\) and let \(\varepsilon>0\). There is \(K(\varepsilon)>1\) so that \(\| S_{\text{rel}}(s)\|<\varepsilon\) implies that \(\Gamma_2\) is a \(K(\varepsilon)\)-quasiconformal deformation of \(\Gamma_1\), where \(K(\varepsilon)\to 1\) as \(\varepsilon\to 0\).
The following two results are corollaries of the main theorem. The first result was obtained early by P. Perry [Ann. Acad. Sci. Fenn., Ser A I, Math. 20, No. 2, 251-257 (1995; Zbl 0861.57017)]. Suppose that the conditions of the Main Theorem are satisfied and \(S_{\text{rel}}(s)=0\). Then the manifolds \(M(\Gamma_1)\) and \(M(\Gamma_2)\) are isometric.
The second one was motivated by the results of F. W. Gehring and J. Väisälä [J. Lond. Math. Soc., II. Ser. 6, 504-512 (1973; Zbl 0258.30020)] and K. Astala [Acta Math. 173, No. 1, 37-60 (1994; Zbl 0815.30015)]. For a set \(E\subset\widehat{\mathbb{C}}\) let \(D(E)\) be the Hausdorff dimension of \(E\). Suppose \(\Gamma_1\) and \(\Gamma_2\) are convex cocompact groups satisfying the conditions given in the Main Theorem. Then there exists a \(\nu(\varepsilon)>0\) so that \(| D(L_{\Gamma_1})-D(L_{\Gamma_2})|<\nu(\varepsilon)\), where \(\nu(\varepsilon)\to 0\) as \(\varepsilon\to 0\).

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

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