27 research outputs found

    Conformal grafting and convergence of Fenchel-Nielsen twist coordinates

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    We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge.Comment: 18 pages, 4 figure

    The converse of the Schwarz Lemma is false

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    Let h : X β†’ Y be a homeomorphism between hyperbolic surfaces with finite topology. If h is homotopic to a holomorphic map, then every closed geodesic in X is at least as long as the corresponding geodesic in Y, by the Schwarz Lemma. The converse holds trivially when X and Y are disks or annuli, and it holds when X and Y are closed surfaces by a theorem of Thurston. We prove that the converse is false in all other cases, strengthening a result of Masumoto

    Failure of the well-rounded retract for Outer space and Teichm\"uller space

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    The well-rounded retract for SLn(Z)\mathrm{SL}_n(\mathbb{Z}) is defined as the set of flat tori of unit volume and dimension nn whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space XnX_n of metric graphs of rank nn and the Teichm\"uller space Tg\mathcal{T}_g of closed hyperbolic surfaces of genus gg, we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the set of graphs whose systoles fill (the analogue of a set proposed as a spine for Tg\mathcal{T}_g by Thurston) is a spine for XnX_n but that its dimension is larger than the virtual cohomological dimension of Out(Fn)\mathrm{Out}(F_n) in general.Comment: 8 page

    The dimension of Thurston's spine

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    We show that for every Ξ΅>0\varepsilon>0, there exists some gβ‰₯2g\geq 2 such that the set of closed hyperbolic surfaces of genus gg whose systoles fill has dimension at least (5βˆ’Ξ΅)g(5-\varepsilon) g. In particular, the dimension of this set -- proposed as a spine for moduli space by Thurston -- is larger than the virtual cohomological dimension of the mapping class group.Comment: Updated to published versio