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arXiv · 2303.11481

Bianchi and Hilbert-Blumenthal quaternionic orbifolds

Alberto Verjovsky·Adrian Zenteno
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Abstract

In a series of papers, published in Mathematische Annalen, Bianchi and Blumenthal introduced the notions of Bianchi orbifolds and Hilbert-Blumnethal surfaces as generalizations of modular curves associated to quadratic fields. In this paper, in the same spirit, and following a similar line of reasoning, we introduce the concept of Bianchi and Hilbert-Blumenthal quaternionic orbifolds as generalizations of the Lipschitz and Hurwitz quaternionic modular orbifolds defined recently by D\'iaz, Vlacci and the first author. In particular, we describe the cusp cross-sections of the Hilbert-Blumenthal quaternionic orbifolds in terms of fundamental units of real quadratic fields. These are 7-dimensional solvmanifolds which are virtual ${\mathbb T}^6$ bundles over the circle with monodromy a linear Anosov diffeomorphism of the 6-torus.

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Alberto Verjovsky, Adrian Zenteno. 2023-03-20. Bianchi and Hilbert-Blumenthal quaternionic orbifolds. https://arxiv.org/abs/2303.11481

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