Hyperbolic limits of cantor set complements in the sphere

Abstract

Let 𝑀 be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in 𝕊 3 . Then, if 𝑀 admits an exhaustion by 𝜋 1 -injective sub-manifolds there exists Cantor sets 𝐶 𝑛 ⊆ 𝕊 3 such that 𝑁 𝑛 = 𝕊 3 ⧵ 𝐶 𝑛 is hyperbolic and 𝑁 𝑛 → 𝑀 geometrically.M S C ( 2 0 2 0 ) 30F40 (primary)

Effective contraction of Skinning maps

“…Indeed, consider all manifolds M ∈ M B such that for all i ∈ N every component U i := M i \ M i−1 is acylindrical. By the main results of [6] this guarantees that M is in fact hyperbolic, which is in general not the case, see [5,7], or [8,9] for other examples of infinite-type hyperbolic 3-manifolds. We can thus think of a (hyperbolic) metric g on M as a gluing of (hyperbolic) metrics g i on the U i 's and so it makes sense to investigate the glueing of pairs U i , U i+1 via skinning maps.…”

Untitled

“…Indeed, consider all manifolds M ∈ M B such that for all i ∈ N every component U i := M i \ M i−1 is acylindrical. By the main results of [6] this guarantees that M is in fact hyperbolic, which is in general not the case, see [5,7], or [8,9] for other examples of infinite-type hyperbolic 3-manifolds. We can thus think of a (hyperbolic) metric g on M as a gluing of (hyperbolic) metrics g i on the U i 's and so it makes sense to investigate the glueing of pairs U i , U i+1 via skinning maps.…”