Nearly Fuchsian surface subgroups of finite covolume Kleinian groups

On the degree of a finite cover which fibers over circle

“…At first, by [KW21], any cusped hyperbolic 3-manifold M contains a π 1 -injective quasi-Fuchsian closed subsurface S M . If M has only one cusp (e.g.…”

The panted cobordism group of cusped hyperbolic 3-manifolds

“…More precisely, Kahn and Markovic used tools in geometry and dynamics to paste a lot of (R, )-good pants in M along (R, )-good curves in a nearly geodesic manner, to get the desired π 1 -injective subsurface (see Section 2.1 for definitions). This is the starting point of a collection of works under the theme of good pants construction, which use various versions of good pants to construct π 1 -injective subsurfaces in geometrically interesting spaces ([KM1,KM2,LM,Ham,KW1]).…”

“…All the above results on good pants construction only work for closed hyperbolic 3-manifolds, since good pants are not equidistributed along good curves in cusped hyperbolic 3-manifolds. In [KW1], Kahn and Wright overcame this difficulty, and generalized Kahn and Markovic's surface subgroup theorem to cusped hyperbolic 3-manifolds. Besides (R, )-good pants, they introduced a new object called (R, )good hamster wheel, as another building block of the π 1 -injective subsurface.…”

“…Besides (R, )-good pants, they introduced a new object called (R, )good hamster wheel, as another building block of the π 1 -injective subsurface. Our paper will not use the main result in [KW1]. However, we will use some geometric ideas in [KM1] and adopt some basic notions in that paper, including a height function in cusped hyperbolic 3-manifolds.…”

“…The "moreover" part of Theorem 1.1 is a nice and neat result by itself. However, in order to study virtual domination of 3-manifolds in a forthcoming paper [Sun], we need to adopt the construction in [KW1], which requires the result in Theorem 1.1 on Ω α ln R,β ln R R, (M ). Given Theorem 1.1, it is natural to ask the following question on the panted cobordism group with no height control.…”