Abstract
We develop a general representation theoretic framework for trace formulas for quotients of rank one simple Lie groups by convex-cocompact discrete subgroups. We further discuss regularized traces of resolvents with applications to Selberg-type zeta functions.
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Notes
- 1.
The problem cannot be solved by just looking at a space of functions f having not necessarily compact support to which Ψ can be applied such that Lemma 1 remains valid. If the critical exponent of Γ is positive, any reasonable space of this kind is contained in some L p(G) for p < 2. The only L p-functions f on G (p < 2) with compactly supported Fourier transform are linear combinations of matrix coefficients of certain discrete series representations. They satisfy Ψ(f) = 0, compare Proposition 3.
- 2.
The notion of measurability is part of the structure of the direct integral, see e.g. [22], Chap. 14. In our case it just amounts to the measurable dependence on the inducing parameters discussed in the previous section.
- 3.
Compare the footnote on page 5.
- 4.
Except for some minor corrections and additions including more recent references.
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Acknowledgements
This paper has been writtenFootnote 4 in the years 1999/2000. It is an outcome of the exciting research environment of the Göttingen Mathematical Institute which was heavily stimulated by the interests and mathematical influence of Samuel J. Patterson.
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Bunke, U., Olbrich, M. (2012). Towards the Trace Formula for Convex-Cocompact Groups. In: Blomer, V., Mihăilescu, P. (eds) Contributions in Analytic and Algebraic Number Theory. Springer Proceedings in Mathematics, vol 9. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1219-9_5
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