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Farre, James ; Vargas Pallete, Franco ( , Proceedings of the American Mathematical Society)By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic 3 3 manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic 3 3 manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.more » « less

Farre, James ( , International Mathematics Research Notices)Abstract We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is nonzero in degree three bounded cohomology and has maximal seminorm, for any discrete group $G$. When $n=2$, the Borel class is equal to the threedimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in seminorm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a nonabelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce nontrivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.more » « less

Farre, James ( , Journal of Topology)