Geometric Zeta Functions, $L^2$-Theory, and Compact Shimura Manifolds

Details Geometric Zeta Functions, $L^2$-Theory, and Compact Shimura Manifolds Geometric Zeta Functions, $L^2$-Theory, and Compact Shimura Manifolds

1995-03-07

arxiv.org/abs/math/9503216v1

Mathematics

Differential Geometry

Scientific paper

We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray and Singer and the holomorphic $L^2$-torsion, where the latter is defined via the $L^2$-theory of Atiyah. For higher fundamental rank twisted torsion numbers appear.

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