(ℤ₂)³-colorings and right-angled hyperbolic 3-manifolds

Youlin Li and Jiming Ma

Vol. 263 (2013), No. 2, 419–434

DOI: 10.2140/pjm.2013.263.419

Abstract

For a compact 3-manifold N with connected nonempty boundary, let Γ be an admissible trivalent graph in ∂N that decomposes ∂N into a set of disks. As an extension of small covers, from a (ℤ₂)³-coloring λ on ∂N − Γ, one can get a closed 3-manifold M_λ that admits a locally standard (ℤ₂)³-action.

Suppose N is irreducible and atoroidal: say, a handlebody. We give a combinatorial necessary and sufficient condition for a (ℤ₂)³-colorable pair (N,Γ) to admit a right-angled hyperbolic structure, which naturally induces a hyperbolic structure on M_λ.

Keywords

(ℤ₂)³-action, hyperbolic structure with polyhedral boundary, 3-manifold

Mathematical Subject Classification 2010

Primary: 57M50, 57M60

Secondary: 52B70

Milestones

Received: 31 March 2012

Revised: 11 November 2012

Accepted: 25 November 2012

Published: 31 May 2013