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AN INFINITE FAMILY OF HYPERBOLIC GRAPH COMPLEMENTS IN S³

R. FRIGERIO

Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo, 5, 56127 Pisa, Italy

https://doi.org/10.1142/S0218216505003919Cited by:3 (Source: Crossref)

Abstract

For any g ≥ 2 we construct a graph Γ_g ⊂ S³ whose exterior M_g = S³\N(Γ_g) supports a complete finite-volume hyperbolic structure with one toric cusp and a connected geodesic boundary of genus g. We compute the canonical decomposition and the isometry group of M_g, showing in particular that any self-homeomorphism of M_g extends to a self-homeomorphism of the pair (S³,Γ_g), and that Γ_g is chiral. Building on a result of Lackenby we also show that any non-meridinal Dehn filling of M_g is hyperbolic, thus getting an infinite family of graphs in S² × S¹ whose exteriors support a hyperbolic structure with geodesic boundary.

Keywords:

AMSC: Primary 57M50, Secondary 57M15

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