| THE MINIMAL ENTROPY PROBLEM FOR 3-MANIFOLDS WITH ZERO SIMPLICIAL VOLUME (2008) | |||||||||||||
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| Abstract. In this note, we consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal (topological) entropy, for certain classes of closed 3-manifolds. Specifically, we prove the following two results. Theorem A. Let M be a closed orientable irreducible 3-manifold whose fundamental group contains a Z ⊕ Z subgroup. The following are equivalent: (1) the simplicial volume �M � of M is zero and the minimal entropy problem for M can be solved; (2) M admits a geometric structure modelled on E 3 or Nil; (3) M admits a smooth metric g with htop(g) = 0. Theorem B. Let M be a closed orientable geometrizable 3-manifold. The following are equivalent: (1) the simplicial volume �M � of M is zero and the minimal entropy problem for M can be solved; (2) M admits a geometric structure modelled on S 3, S 2 × R, E 3, or Nil; (3) M admits a smooth metric g with htop(g) = 0. | |||||||||||||
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