FREE ACTIONS OF FINITE ABELIAN GROUPS ON 3-DIMENSIONAL NILMANIFOLDS

Title & Authors
FREE ACTIONS OF FINITE ABELIAN GROUPS ON 3-DIMENSIONAL NILMANIFOLDS
Choi, Dong-Soon; Shin, Joon-Kook;

Abstract
We study free actions of finite abelian groups on 3­dimensional nilmanifolds. 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. All such actions are completely classified.
Keywords
group actions;Heisenberg group;almost Bieberbach groups;Affine conjugacy;
Language
English
Cited by
1.
Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group, Journal of Topology, 2015, 8, 3, 842
2.
Group extensions and free actions by finite groups on solvmanifolds, Mathematische Nachrichten, 2010, n/a
References
1.
H. Y. Chu and J. K. Shin, Free actions of finite groups on the 3-dimensional nilmanifold, Topology Appl. 144 (2004), 255-270

2.
K. Dekimpe, P. Igodt, S. Kim, and K. B. Lee, Affne structures for closed 3-dimensional manifolds with nil-geometry, Quart. J. Mech. Appl. Math. 46 (1995), 141-167

3.
K. Y. Ha, J. H. Jo, S. W. Kim, and J. B. Lee, Classification of free actions of finite groups on the 3-torus, Topology Appl. 121 (2002), no. 3, 469-507

4.
W. Heil, On $P^2$-irreducible 3-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772- 775

5.
W. Heil, Almost sufficiently large Seifert fiber spaces, Michigan Math. J. 20 (1973), 217-223

6.
J. Hempel, Free cyclic actions of $S^1{\times}S^1{\times}S^1$, Proc. Amer. Math. Soc. 48 (1975), no. 1, 221-227

7.
K. B. Lee, There are only finitely many infra-nilmanifolds under each manifold, Quart. J. Mech. Appl. Math. 39 (1988), 61-66

8.
K. B. Lee and F. Raymond, Rigidity of almost crystallographic groups, Contemp. Math. 44 (1985), 73-78

9.
K. B. Lee, J. K. Shin, and Y. Shoji, Free actions of finite abelian groups on the 3-Torus, Topology Appl. 53 (1993), 153-175

10.
P. Orlik, Seifert Manifolds, Lecture Notes in Math. 291, Springer-Verlag, Berlin, 1972

11.
P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-489

12.
J. K. Shin, Isometry groups of unimodular simply connected 3-dimensional Lie groups, Geom. Dedicata 65 (1997), 267-290

13.
F. Waldhausen, On irreducible 3-manifolds which are suffciently large, Ann. of Math. 87 (1968), no. 2, 56-88

14.
S. Wolfram, Mathematica, Wolfram Research, 1993