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arXiv · 2305.04996

A first Kronecker limit formula for Kleinian groups

Zihan Miao·Anh Nguyen·Tian An Wong
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Abstract

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.

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Zihan Miao, Anh Nguyen, Tian An Wong. 2023-05-08. A first Kronecker limit formula for Kleinian groups. https://arxiv.org/abs/2305.04996

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