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
Abstract

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 ${ℍ}^{4}$, the complex hyperbolic plane ${ℍ}_{ℂ}^{2}$ or the product of two hyperbolic planes ${ℍ}^{2}×{ℍ}^{2}$ then $M$ admits an $\mathsc{ℱ}$–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.

Keywords
minimal entropy, geodesic flows, geometric structures
Mathematical Subject Classification 2000
Primary: 37B40, 57M50
Secondary: 22F30, 53D25