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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.
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 $gamm
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 t
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
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 ex
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.