A first Kronecker limit formula for Kleinian groups

We prove a first Kronecker limit formula for cofinite discrete subgroups of SL\((2,\mathbb{C})\), also called Kleinian groups, generalizing a method of Goldstein over SL\((2,\mathbb R)\). The proof uses the Fourier expansion of Eisenstein series, leading to an analogue of the logarithm of the Dedeki...

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Bibliographic Details
Published in:arXiv.org 2023-05
Main Authors: Miao, Zihan, Nguyen, Anh, Tian An Wong
Format: Article
Language:English
Subjects:
Fourier series
Subgroups
Online Access:Get full text
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Summary:We prove a first Kronecker limit formula for cofinite discrete subgroups of SL\((2,\mathbb{C})\), also called Kleinian groups, generalizing a method of Goldstein over SL\((2,\mathbb R)\). The proof uses the Fourier expansion of Eisenstein series, leading to an analogue of the logarithm of the Dedekind eta function, and whose transformation law produces a function analogous to one related to Dedekind sums. We derive several basic properties of this new function.
ISSN:2331-8422