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Register a new userThe topological types of length bounded multicurves
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Authors
Hugo Parlier
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This article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality that holds over all of moduli space.
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Let $gamma_0$ be a curve on a surface $Sigma$ of genus $g$ and with $r$ boundary components and let $pi_1(Sigma)curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $gamma$ of type $gamma_0$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$ of $pi_1(Sigma)$ the limit $$lim_{Ltoinfty}frac 1{L^{6g-6+2r}}{gammatext{ of type }gamma_0text{ with }Stext{-translation length}le L}$$ exists and is positive. The main new technical tool is that the function which associates to each curve its stable length with respect to the action on $X$ extends to a (unique) continuous and homogenous function on the space of currents. We prove that this is indeed the case for any action of a torsion free hyperbolic group.
A textit{multicurve} $C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{C}$ about $C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$ such that $h^n = t_{C}$. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for $n$. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4.
In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard $(mathbb{Z}_2)^3$-action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional small covers. These six operations are interesting because of their combinatorial natures. Then we show that each 3-dimensional small cover can be obtained from $mathbb{R}P^3$ and $S^1timesmathbb{R}P^2$ with certain $(mathbb{Z}_2)^3$-actions under these six operations. As an application, we classify all 3-dimensional small covers up to $({Bbb Z}_2)^3$-equivariant unoriented cobordism.
Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular homology (on the category of CW-complexes), and it was then shown by Loeh that, in the absolute case, such isomorphism is in fact an isometry with respect to the L^1-seminorm on singular homology and the total variation seminorm on measure homology. Loehs result plays a fundamental role in the use of measure homology as a tool for computing the simplicial volume of Riemannian manifolds. This paper deals with an extension of Loehs result to the relative case. We prove that relative singular homology and relative measure homology are isometrically isomorphic for a wide class of topological pairs. Our results can be applied for instance in computing the simplicial volume of Riemannian manifolds with boundary. Our arguments are based on new results about continuous (bounded) cohomology of topological pairs, which are probably of independent interest.
We show there exist tunnel number one hyperbolic 3-manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams.
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