Igor Nikolaev

Noncommutative geometry of hyperbolic 3-manifolds


Let $M$ be surface bundle over the circle with a pseudo-Anosov monodromy $\phi$. Thurston showed that $M$ is a hyperbolic 3-manifold. One can associate to $M$ an ordered abelian group $E$ (a.k.a ``dimension group'') coming from a $\phi$-invariant geodesic lamination. We prove that $E$ is dimension group of a stationary type. Intrinsically, group $E$ is described by its endomorphism ring. We calculate $End~E\simeq O_K$, where $O_K$ is the ring of integers of a quadratic number field $K=Q(\sqrt{d})$. {\bf Theorem 1} Let $M$ be hyperbolic manifold, which has minimal volume in its commensurability class. Then $Vol~M=2\log\varepsilon/\sqrt{d}$, where $\varepsilon$ is the fundamental unit of $K$. {\bf Theorem 2} Let $h_K$ be the class number of field $K$. Then Gromov-Thurston map $M\mapsto Vol~M$ has degree $h_K$ in the point $M$. Both theorems are proved by the methods of noncommutative geometry [1]. References: [1] I.Nikolaev, K-theory of hyperbolic 3-manifolds, math.GT/0110227.

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