Projective structures with degenerate holonomy and the Bers density conjecture.

*(English)*Zbl 1137.30014Let \(M = S\times [-1,1]\) be an \(I\)-bundle over a closed surface \(S\) of genus \(> 1\). The author is interested in the space \(AH(S)\) of all Kleinian groups isomorphic to \(\pi_1(S)\). By a theorem of Bonahon, this is equivalent to studying complete hyperbolic structures on the interior of \(M\). A generic hyperbolic structure on \(M\) is quasi-fuchsian and the geometry is well understood outside of a compact set. In particular, although the geometry of the surfaces \(S\times\{t\}\) will grow exponentially as \(t\) limits to \(-1\) or \(1\), the conformal structures will stabilize and limit to Riemann surfaces \(X\) and \(Y\). Then \(M\) can be conformally compactified by viewing \(X\) and \(Y\) as conformal structures on \(S\times\{-1\}\) and \(S\times \{1\}\), respectively. Bers showed that \(X\) and \(Y\) parametrize the space \(\text{QF}(S)\) of all quasi-Fuchsian structures. In other words \(\text{QF}(S)\) is isomorphic to \(T(S)\times T(S)\) where \(T(S)\) is the Teichmüller space of marked conformal structures on \(S\).

Let the Bers slice \(B_X\) be the slice of \(\text{QF}(S)\) obtained by fixing \(X\) and letting \(Y\) vary in \(T(S)\). This gives an interesting model of \(T(S)\) because \(B_X\) naturally embeds as a bounded domain in the space \(P(X)\) of projective structures on \(S\) with conformal structure \(X\). The closure \({\overline B}_X\) of \(B_X\) in \(P(X)\) is then a compactification of Teichmüller space. Bers made the following conjecture, known as Bers density conjecture: Let \(\Gamma\in AH(S)\) be a Kleinian group. If \(M =\mathbb H^3/\Gamma\) is singly degenerate then \(\Gamma\in {\overline B}_X\), where \(X\) is the conformal boundary of \(M\) [L. Bers, Ann. Math. 91, 570–600 (1970; Zbl 0197.06001)]. There are some special cases where the conjecture is known. Abikoff proved the conjecture when \(M\) is geometrically finite. Recently Y. N. Minsky [Invent. Math. 146, No. 1, 143–192 (2001; Zbl 1061.37026)] has proved the conjecture in the case where there is a lower bound on the length of any closed geodesic in \(M\) and \(\Gamma\) has no parabolics (\(M\) has bounded geometry). Minsky also proved the conjecture if \(S\) is a punctured torus [Y. N. Minsky, Ann. Math. (2) 149, No. 2, 559–626 (1999; Zbl 0939.30034)].

In this paper, the author proves the conjecture when \(M\) has a sequence of closed geodesies \(c_i\) whose length limits to zero (\(M\) has unbounded geometry). Combined with Minsky’s result the author has an almost complete resolution of Bers’s conjecture: Assume that \(\Gamma\in AH(S)\) has no parabolics. If \(M =\mathbb H^3/\Gamma\) is singly degenerate then \(\Gamma\in {\overline B}_X\), where \(X\) is the conformal boundary of \(M\).

Let the Bers slice \(B_X\) be the slice of \(\text{QF}(S)\) obtained by fixing \(X\) and letting \(Y\) vary in \(T(S)\). This gives an interesting model of \(T(S)\) because \(B_X\) naturally embeds as a bounded domain in the space \(P(X)\) of projective structures on \(S\) with conformal structure \(X\). The closure \({\overline B}_X\) of \(B_X\) in \(P(X)\) is then a compactification of Teichmüller space. Bers made the following conjecture, known as Bers density conjecture: Let \(\Gamma\in AH(S)\) be a Kleinian group. If \(M =\mathbb H^3/\Gamma\) is singly degenerate then \(\Gamma\in {\overline B}_X\), where \(X\) is the conformal boundary of \(M\) [L. Bers, Ann. Math. 91, 570–600 (1970; Zbl 0197.06001)]. There are some special cases where the conjecture is known. Abikoff proved the conjecture when \(M\) is geometrically finite. Recently Y. N. Minsky [Invent. Math. 146, No. 1, 143–192 (2001; Zbl 1061.37026)] has proved the conjecture in the case where there is a lower bound on the length of any closed geodesic in \(M\) and \(\Gamma\) has no parabolics (\(M\) has bounded geometry). Minsky also proved the conjecture if \(S\) is a punctured torus [Y. N. Minsky, Ann. Math. (2) 149, No. 2, 559–626 (1999; Zbl 0939.30034)].

In this paper, the author proves the conjecture when \(M\) has a sequence of closed geodesies \(c_i\) whose length limits to zero (\(M\) has unbounded geometry). Combined with Minsky’s result the author has an almost complete resolution of Bers’s conjecture: Assume that \(\Gamma\in AH(S)\) has no parabolics. If \(M =\mathbb H^3/\Gamma\) is singly degenerate then \(\Gamma\in {\overline B}_X\), where \(X\) is the conformal boundary of \(M\).

Reviewer: Vasily A. Chernecky (Odessa)

##### MSC:

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

30F60 | Teichmüller theory for Riemann surfaces |