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We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an immersed surface. In the case that the orbifold is a disk, there are some conditions. Our results generalize work of Calegari-Louwsma and resolve a conjecture of Calegari.
An immersion of a smooth $n$-dimensional manifold $M to mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the minimum dimens
In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double
We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed
We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichm{u}ller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we com
We show that Mirzakhanis curve counting theorem also holds if we replace surfaces by orbifolds.