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Analytic Torsion, Dynamical Zeta Function, and the Fried Conjecture for Admissible Twists

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Abstract

We show an equality between the analytic torsion and the absolute value at zero of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author’s previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.

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Notes

  1. The case of even dimension is trivial [Sh19, Remark 5.12] (c.f. Remark 4.5).

  2. By Margulis’ super-rigidity [M91, Section VII.5] (see also [BoW00, Section XIII.4.6]), this is the most interesting case, when the real rank of the locally symmetric space is \(\geqslant 2\).

  3. If G is semisimple or more generally if G has a compact centre, then all the representations of G has an admissible metric ([MatMu63, Lemma 3.1], Proposition 2.9).

  4. More precisely, we need assume that the Casimir of \(\mathfrak {g}\) acts on \(\rho \) as a scalar.

  5. By [BSh19, Theorem 2.3], this is indeed independent of the choice of \(g_{\gamma }\).

  6. The quantity \(\ell _{[\gamma ]}\) defined in (2.16) depends only on the conjugacy class of \(\gamma \) in G. So they are well defined on the conjugacy classes of \(\Gamma \).

  7. A more general construction for the Selberg zeta function is given in [Sh20], which is associated to \(\eta \) and to an arbitrary representation of \(\rho :\Gamma \rightarrow {{\,\mathrm{\mathrm GL}\,}}_{r}(\mathbf {C})\).

  8. They are called Dirac cohomology of E (see [HuPa06]).

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  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, 4 Place Jussieu, 75252, Paris Cedex 5, France

    Shu Shen

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The author acknowledges the partial support by the Grant ANR-20-CE40-0017.

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Shen, S. Analytic Torsion, Dynamical Zeta Function, and the Fried Conjecture for Admissible Twists. Commun. Math. Phys. 387, 1215–1255 (2021). https://doi.org/10.1007/s00220-021-04113-y

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