The Taylor expansion of Ruelle L-function at the origin and the Borel regulator

Details The Taylor expansion of Ruelle L-function at the origin and the Borel regulator The Taylor expansion of Ruelle L-function at the origin and the Borel regulator

2008-04-17

arxiv.org/abs/0804.2715v1

Mathematics

Differential Geometry

50 pages, 0 figures

Scientific paper

We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rational multiple of the volume up to a certain contribution from cusps. Moreover if the dimension is three we will identify the leading coefficient. Both of them will be intepreted by the Borel regulator in algebraic K-theory. Also a relation with the L^2-torsion will be discussed.

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