On the uniqueness conjecture for Markoff triples
This paper has been withdrawn by author due to an error in the proof.
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arXiv subjects
Qing Zhou
Publications and source records attributed to Qing Zhou.
On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds
For any hyperbolic 3-manifold $M$ with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of $\partial M$ or the volume of $M$ is bounded above. When the volume is bounded above, then area of $\partial M$ is bounded above and the length of closed geodesic on $\partial M$ is bounded below.
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Generalized Hopfian property, minimal Haken manifold, and J. Simon's conjecture for 3-manifold groups
We address a conjecture that $π_1$-surjective maps between closed aspherical 3-manifolds having the same rank on $π_1$ must be of non-zero degree. The conjecture is proved for Seifert manifolds, which is used in constructing the first known example of minimum Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds.
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Any 3-manifold 1-dominates at most finitely many geometric 3-manifolds
In this paper, we proved that any closed orientable 3-manifold 1-dominates at most finitely many geometric 3-manifolds.
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The Moduli Space of Hyperbolic Cone Structures
Let $Σ$ be a hyperbolic link with $m$ components in a 3-dimensional manifold $X$. In this paper, we will show that the moduli space of marked hyperbolic cone structures on the pair $(X, Σ)$ with all cone angle less than $2π/3$ is an $m$-dimensional open cube, parameterized naturally by the $m$ cone angles. As a corollary, we will give a proof of a special case of Thurston's geometrization theorem for orbifolds.
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Pseudo-Anosov maps and simple closed curves on surfaces
Given a pair of curves C_1 and C_2 on a hyperbolic surface F, when does there exist a pseudo-Anosov map sending one to another? More generally, one may ask the same question for C_i to be sets of disjoint simple closed curves. We will give necessary and sufficient conditions for the existence of such maps.
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