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FINITE GROUP ACTIONS ON THE 3–DIMENSIONAL NILMANIFOLD
Daehwan Goo,Joonkook Shin 충청수학회 2005 충청수학회지 Vol.18 No.2
We study only free actions of finite groups G on the 3-dimensional nilmanifold, up to topological conjugacy which yields an infra-nilmanifold of type 2.
FREE ACTIONS OF FINITE GROUPS ON 3-DIMENSIONAL NILMANIFOLDS WITH HOMOTOPICALLY TRIVIAL TRANSLATIONS
Daehwan Koo,Eunmi Park,Joonkook Shin 충청수학회 2020 충청수학회지 Vol.33 No.1
affine conjugacy;almost Bieberbach group;group action;Heisenberg group;homotopically trivial translation
FREE CYCLIC ACTIONS OF THE 3-DIMENSIONAL NILMANIFOLD
Shin, Joonkook,Goo, Daehwan,Park, Eunmi 충청수학회 2001 충청수학회지 Vol.14 No.2
We shall deal with ten cases out of 15 distinct almost Bieberbach groups up to Seifert local invariant. In those cases we will show that if G is a finite abelian group acting freely on the standard nilmanifold, then G is cyclic, up to topological conjugacy.
NONABELIAN GROUP ACTIONS ON 3-DIMENSIONAL NILMANIFOLDS REVERSING FIBER ORIENTATION
Koo, Daehwan,Lee, Taewoong,Shin, Joonkook Chungcheong Mathematical Society 2018 충청수학회지 Vol.31 No.4
We study free actions of finite nonabelian groups on 3-dimensional nilmanifolds with the first homology ${\mathbb{Z}}^2{\bigoplus}{\mathbb{Z}}_2$ which yield an orbit manifold reversing fiber orientation, up to topological conjugacy. We show that those nonabelian groups are $D_4$(the dihedral group), $Q_8$(the quaternion group), and $C_8.C_4$(the $1^{st}$ non-split extension by $C_8$ of $C_4$ acting via $C_4/C_2=C_2$).
CLASSIFICATION OF FREE ACTIONS OF FINITE GROUPS ON 3-DIMENSIONAL NILMANIFOLDS
Koo, Daehwan,Oh, Myungsung,Shin, Joonkook Korean Mathematical Society 2017 대한수학회지 Vol.54 No.5
We study free actions of finite groups on 3-dimensional nil-manifolds with the first homology ${\mathbb{Z}}^2{\oplus}{\mathbb{Z}}_p$. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal nilpotent subgroups of almost Bieberbach groups of finite index, up to affine conjugacy.