Dehn surgery, homology and hyperbolic volume

Abstract

If a closed, orientable hyperbolic 3--manifold M has volume at most 1.22 then H_1(M;Z_p) has dimension at most 2 for every prime p not 2 or 7, and H_1(M;Z_2) and H_1(M;Z_7) have dimension at most 3. The proof combines several deep results about hyperbolic 3--manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic C in M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C.

This is the version published by Algebraic & Geometric Topology on 8 December 2006

References in corpus (3)

• Tameness of hyperbolic 3-manifolds
• Lower bounds on volumes of hyperbolic Haken 3-manifolds
• Volume change under drilling

Cited by in corpus (17)

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• Hyperbolic 3-manifolds of low cusp volume
• Small volume link orbifolds
• New Lower Bounds for Optimal Horoball Packing Density in Hyperbolic $n$-space for $6\le n\le 9$
• The lowest volume 3-orbifolds with high torsion
• Rank gradient of small covers
• Small optimal Margulis numbers force upper volume bounds
• Small isospectral and nonisometric orbifolds of dimension 2 and 3
• Volume and homology of one-cusped hyperbolic 3-manifolds
• Margulis numbers for Haken manifolds
• Optimal Horoball Packing Densities for Koszul-type tilings in Hyperbolic $3$-space
• Volume and Homology for Hyperbolic $3$-Orbifolds, I
• Margulis numbers and number fields