The Minimal Volume Orientable Hyperbolic 3-manifold with 4 Cusps
نویسندگان
چکیده
We prove that the 82 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V8 of the ideal regular octahedron, i.e. 7.32... = 2V8. The proof relies on Agol’s argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also need to estimate the volume of a hyperbolic 3-manifold with totally geodesic boundary which contains an essential surface with non-separating boundary.
منابع مشابه
The Minimal Volume Orientable Hyperbolic 2-cusped 3-manifolds
We prove that the Whitehead link complement and the (−2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 × Catalan’s constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.
متن کاملCompact Hyperbolic 4-manifolds of Small Volume
We prove the existence of a compact non-orientable hyperbolic 4-manifold of volume 32π2/3 and a compact orientable hyperbolic 4-manifold of volume 64π2/3, obtainable from torsion-free subgroups of small index in the Coxeter group [5, 3, 3, 3]. At the time of writing these are the smallest volumes of any known compact hyperbolic 4-manifolds.
متن کاملAll finitely presentable groups from link complements and Kleinian groups
Klein defined geometry in terms of invariance under groups actions; here we give a discrete (partial) converse of this, interpreting all (finitely presentable) groups in terms of the geometry of hyperbolic 3-manifolds (whose fundamental groups are, appropriately, Kleinian groups). For G∗ a Kleinian group of isometries of hyperbolic 3-space H, with MG∗ ∼= H3/G∗ a non-compact N -cusped orientable...
متن کاملFour-free Groups and Hyperbolic Geometry
We give new geometric information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that vol M ≤ 3.44, then dimZ2 H1(M ; Z2) ≤ 7.
متن کاملThe Volume of a Hyperbolic 3-manifold with Betti Number 2
In a forthcoming paper [4] we will show that if one excludes certain special manifolds, such as fiber bundles over S, then the same estimate holds for hyperbolic manifolds with Betti number 1. Our volume estimates can be placed in context by comparing them with volume estimates for general hyperbolic manifolds, as well as with volumes of known examples. The largest known lower bound for the vol...
متن کامل