TY - JOUR

T1 - On the coverings of Hantzsche-Wendt manifold

AU - Chelnokov, Grigory

AU - Mednykh, Alexander

N1 - Funding Information: 2010 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. ∗The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. 0314-2019-0007). Publisher Copyright: © 2022 Tohoku University, Mathematical Institute. All rights reserved.

PY - 2022

Y1 - 2022

N2 - There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable G1, . . . , G6 and four are non-orientable B1, . . . , B4. In the present paper we investigate the manifold G6, also known as Hantzsche-Wendt manifold; this is the unique Euclidean 3-form with finite first homology group H1(G6) = Z24 . The aim of this paper is to describe all types of n-fold coverings over G6 and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group π1(G6) 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.

AB - There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable G1, . . . , G6 and four are non-orientable B1, . . . , B4. In the present paper we investigate the manifold G6, also known as Hantzsche-Wendt manifold; this is the unique Euclidean 3-form with finite first homology group H1(G6) = Z24 . The aim of this paper is to describe all types of n-fold coverings over G6 and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group π1(G6) 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.

KW - crystallographic

KW - Euclidean form

KW - flat 3-manifold

KW - non-equivalent coverings

KW - platycosm

UR - http://www.scopus.com/inward/record.url?scp=85135184687&partnerID=8YFLogxK

U2 - 10.2748/tmj.20210308

DO - 10.2748/tmj.20210308

M3 - Article

AN - SCOPUS:85135184687

VL - 74

SP - 313

EP - 327

JO - Tohoku Mathematical Journal

JF - Tohoku Mathematical Journal

SN - 0040-8735

IS - 2

ER -