A locally hyperbolic 3-manifold that is not homotopy equivalent to any hyperbolic 3-manifold

Abstract

We construct a locally hyperbolic 3-manifold M M such that π 1 ( M ) \pi _1(M) has no divisible subgroups. We then show that M M is not homotopy equivalent to any complete hyperbolic manifold.

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]. 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.…”

Hyperbolization of infinite-type 3-manifolds

“…Remark 2.4. By [Cre18a] one can also show that the manifold N is not homotopy equivalent to any hyperbolic 3-manifold.…”

“…However, the 3-manifold M ∞ is homotopy equivalent to a complete hyperbolic 3-manifold. In [Cre18a] we improved the above example by building a 3-manifold N ∈ M B such that N is not homotopy equivalent to any complete hyperbolic 3-manifold.…”

Hyperbolic limits of cantor set complements in the sphere

“…By work of Souto–Stover [31] and Cremaschi–Souto [13] and Cremaschi [10, 12] it is not hard to build hyperbolizable infinite type 3‐manifolds that are homeomorphic to Cantor set complements in the 3‐sphere $$\mathbb {S}^3$$. In particular, in [13], the manifold of Example 2 can be extended to be a Cantor set complement showing, for example, how one can have a hyperbolizable Cantor set complement in $$\mathbb {S}^3$$ whose fundamental group is not residually finite.…”