6,938 research outputs found

    Steiner Variations on Random Surfaces

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    Ambartzumian et.al. suggested that the modified Steiner action functional had desirable properties for a random surface action. However, Durhuus and Jonsson pointed out that such an action led to an ill-defined grand-canonical partition function and suggested that the addition of an area term might improve matters. In this paper we investigate this and other related actions numerically for dynamically triangulated random surfaces and compare the results with the gaussian plus extrinsic curvature actions that have been used previously.Comment: 8 page

    Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces

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    We study threefolds fibred by mirror quartic K3 surfaces. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic K3 surfaces. This is then used to give a complete explicit description of all Calabi-Yau threefolds fibred by mirror quartic K3 surfaces. We conclude by studying the properties of such Calabi-Yau threefolds, including their Hodge numbers and deformation theory.Comment: v2: Significant changes at the request of the referee. Section 3 has been rearranged to accommodate a revised proof of Proposition 3.5 (formerly 3.2). Section 5 has been removed completely, it will instead appear as part of Section 5 in arxiv:1601.0811

    Digraphs and cycle polynomials for free-by-cyclic groups

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    Let \phi \in \mbox{Out}(F_n) be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism ϕ\phi determines a free-by-cyclic group Γ=Fn⋊ϕZ,\Gamma=F_n \rtimes_\phi \mathbb Z, and a homomorphism α∈H1(Γ;Z)\alpha \in H^1(\Gamma; \mathbb Z). By work of Neumann, Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, α\alpha has an open cone neighborhood A\mathcal A in H1(Γ;R)H^1(\Gamma;\mathbb R) whose integral points correspond to other fibrations of Γ\Gamma whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen's Teichm\"uller polynomial that computes the dilatations of all outer automorphism in A\mathcal A.Comment: 41 pages, 20 figure
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