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Let $S$ be a compact orientable 2-surface with $\chi(X)\leq -3$. Y.Minsky and H.Masur proved that the curve complex of $X$ is $\delta$-hyperbolic and infinite. E.Klarreich (see also U.Hamenstadt) prov …

Let $M$ be orientable 3-manifold admitting a Heegaard splitting $V\cup_{S}W$. Let $X$ be a carrier graph of $M$ such that rank($X$)=rank($\pi_{1} M$). Note: A connected graph is called a carrier gr …

Is there an algorithm to detect the Non-Haken Manifold? Or, is there a sufficient condition for a manifold to be a non-Haken manifold? (off course, I hope that condition is not the ones in its def …

Let $H_{1}$ and $H_{2}$ are two handlebodies. If $\partial H_{1}$ and $\partial H_{2}$ are homeomorphic, then $H_{1}\cup_{f} H_{2}$ is a Heegaard splitting. To a general Heegaard splitting, one of $H_ …

Let $M$ be a closed hyperbolic 3-manifold and $H_{g}$ a genus g handlebody. Assume that $\pi: int(H_{g})\rightarrow M$ is a cover. Denote $N\subset H_{g}$ the convex core. My question is: If the diam …

Assume that $M$ is a closed, irreducible, orientable 3-manifold. Suppose that we have a closed, immersed, incompressible surface $F$ of genus at least 1. Since we only required $F$ to be immersed in $ …

I am curious about how the Heegaard genus changes after a finite covering. Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that the Heegaard genus of a finite covering of $N$ …

Suppose $M$ is a closed 3-manifold, $C\subset M$ is a simple closed curve which represent identity in $\pi_{1}(M)$. Then $C$ bounds an immersed disk in $M$. My question is: When does it bound an im …

Assume $V$ is a handlebody and $C$ be a simple closed curve contained in the interior of $V$. As Sam said, there exists some simple closed curve such that every dehn surgery along it produces a handl …

Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold. Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a hyperbolic structure. Thursto …

There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric. M …