All Questions

Question 1: Suppose that two hyperbolic 3-manifolds $M_1$ and $M_2$ with finitely generated fundamental group satisfy the property that for every closed geodesic in $M_1$, there is a closed geodesic ...

There are many results mentioned pairs of pants, and it seems to be a classical model. Why are the pairs of pants so useful? For example, does it have any application if we estimate the perimeter or ...

It is well-known that the set of closed geodesics on a closed hyperbolic surface is dense. My questions: If this property still holds on finite-area hyperbolic surfaces, infinite-area hyperbolic ...

Let $X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $\Gamma < Isom(X)$ a discrete group of ...

Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...