Guts of surfaces in punctured-torus bundles (Q818739)

If \(M\) is a compact 3--manifold and \(S\) a two-sided properly embedded surface in \(M\), \(Guts(M,S)\) is the closure of the complement of the manifold \(M\setminus N(S)\), where \(N(S)\) is a regular neighborhood of \(S\) in \(M\). If \(M\) is a hyperbolic 3--manifold of finite volume and \(S\) and arbitrary incompressible surface in \(M\), Agol proved that \(Vol(M) \geq -2 V_3\chi(Guts (M,S))\), where \(V_3\) denotes the volume of a regular ideal straight 3--simplex in hyperbolic 3--space, and he used this inequality to get useful estimates for the hyperbolic volume of 2--bridge links [I. Agol, \textit{Lower bounds on volumes of hyperbolic Haken manifolds}, Preprint, http://front.math.ucdavis.edu/math. GT/9906182]. In the paper under review the author studies hyperbolic once-punctured torus bundles, where incompressible surfaces are well-understood, and proves that \(Guts(M,S)\) is empty (Theorem 1). Moreover, he gives an upper bound for the hyperbolic volume of a punctured-torus bundle in terms of its monodromy. If \(M\) is a punctured torus bundle with hyperbolic monodromy \(A=t_l^{l_1}t_m^{m_1} \dots t_l^{l_r}t_m^{m_r}\), where \(t_l\) resp. \(t_m\) denotes the Dehn-twist in the longitude resp. meridian of the once-punctured torus, the volume satisfies the following inequality (Teorem 2): \(Vol(M)\leq V_3 \sum_{i=1}^r l_i+m_i.\) The equality holds for finite covers of the figure-eight knot complement.