Tohoku Mathematical Journal
2022
June
SECOND SERIES VOL. 74, NO. 2

Tohoku Math. J.
74 (2022), 313-327

Title ON THE COVERINGS OF HANTZSCHE-WENDT MANIFOLD

Author Grigory Chelnokov and Alexander Mednykh

(Received January 4, 2020, revised March 3, 2021)
Abstract. There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four are non-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. In the present paper we investigate the manifold $\mathcal{G}_6$, also known as Hantzsche-Wendt manifold; this is the unique Euclidean $3$-form with finite first homology group $H_1(\mathcal{G}_6) = \mathbb{Z}^2_4$. The aim of this paper is to describe all types of $n$-fold coverings over $\mathcal{G}_{6}$ and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group $\pi_1(\mathcal{G}_{6})$ up to isomorphism. Given index $n$, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences.

Mathematics Subject Classification. Primary 20H15; Secondary 57M10, 05A15, 55R10.
Key words and phrases. Euclidean form, platycosm, flat 3-manifold, non-equivalent coverings, crystallographic group, Dirichlet generating series, number of subgroups, number of conjugacy classes of subgroups.

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