In: Revista Matemática Complutense, 2015, p. 1–18
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In: Canadian Mathematical Bulletin, 2015, vol. 57, no. 4, p. 870–876
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Seppälä. The goal of this note is to give a short...
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In: Publicacions Matemàtiques, 2012, vol. 57, no. 1, p. 219-237
We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.
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In: Manuscripta Mathematica, 2012, vol. 139, no. 3-4, p. 515-534
A result of Bangert states that the stable norm associated to any Riemannian metric on the 2-torus T ² is strictly convex. We demonstrate that the space of stable norms associated to metrics on T ² forms a proper dense subset of the space of strictly convex norms on R2{/span> . In particular, given a strictly convex norm || · ||∞ on R2{/span> we construct a...
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In: Journal of Topology and Analysis, 2012, vol. 4, no. 3, p. 271
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In: Geometric and functional analysis, 2012, vol. 22, no. 1, p. 37-73
Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions.First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any l Î (0, 1) there exists a constant C λ such that...
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In: Geometriae Dedicata, 2012, vol. 157, no. 1, p. 331-338
The main goal of this note is to show that the study of closed hyperbolic surfaces with maximum length systole is in fact the study of surfaces with maximum length homological systole. The same result is shown to be true for once-punctured surfaces, and is shown to fail for surfaces with a large number of cusps.
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In: Geometric and Functional Analysis, 2011, vol. 21, no. 5, p. 1069-1090
Our goal is to show, in two different contexts, that “random” surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g7/6−ε. Moreover, we prove that this bound holds for most metrics in the modulispace of hyperbolic metrics equipped with the Weil–Petersson volume...
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In: Topology and its Applications, 2011, vol. 158, no. 1, p. 84-92
We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus g which fill and pairwise intersect at most Kgreater-or-equal, slanted1 times is View the MathML source as g→∞. We then bound from below the cardinality of a filling set of systoles by g/log(g). This illustrates that the topological condition that a set of curves...
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