The geometry and topology of arithmetic hyperbolic 3-manifolds
The geometry and topology of arithmetic hyperbolic 3-manifolds
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发表时间：
2007
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通讯作者：
A. Reid
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作者：
A. Reid
and is called the virtual first Betti number of M. Conjecture 1.3. Let M be a closed hyperbolic 3-manifold, then vb1(M) = 1. Conjecture 1.4. Let M be a closed hyperbolic 3-manifold, then 1(M) is large; that is to say, some finite index subgroup of 1(M) admits a surjective homomorphism onto a non-abelian free group. Now it is clear that Conjecture 1.4 implies Conjecture 1.3 implies Conjecture 1.2, and standard 3-manifold topology shows that Conjecture 1.2 implies Conjecture 1.1. Our interest here is in recent work towards reversing these implications. Our geometric discussion is centered around the set of lengths of closed geodesics, as well as the set of geodesics themselves, and in particular on how these force a certain rigidity on commensurability classes. For example, the length spectrum L(M) of a hyperbolic 3-manifold M is the set of all lengths of closed geodesics on M counted with multiplicities. A question that has attracted some attention is whether hyperbolic 3-manifolds with the same length spectra are commensurable. We discuss this and other related questions in this paper.
