Volume 9, issue 1 (2009)

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Minimal entropy and geometric decompositions in dimension four

Pablo Suárez-Serrato

Algebraic & Geometric Topology 9 (2009) 365–395

We show vanishing results about the infimum of the topological entropy of the geodesic flow of homogeneous smooth four-manifolds. We prove that any closed oriented geometric four-manifold has zero minimal entropy if and only if it has zero simplicial volume. We also show that if a four-manifold M admits a geometric decomposition in the sense of Thurston and does not have geometric pieces modelled on hyperbolic four-space 4, the complex hyperbolic plane 2 or the product of two hyperbolic planes 2 × 2 then M admits an –structure. It follows that M has zero minimal entropy and collapses with curvature bounded from below. We then analyse whether or not M admits a metric whose topological entropy coincides with the minimal entropy of M and provide new examples of manifolds for which the minimal entropy problem cannot be solved.

minimal entropy, geodesic flows, geometric structures
Mathematical Subject Classification 2000
Primary: 37B40, 57M50
Secondary: 22F30, 53D25
Received: 21 April 2008
Revised: 5 February 2009
Accepted: 5 February 2009
Published: 25 February 2009
Pablo Suárez-Serrato
Jalisco S/N
Col. Valenciana
CP: 36240 Guanajuato, Gto