UNIMODULAR GROUPS OF TYPE ℝ3 ⋊ ℝ

Title & Authors
UNIMODULAR GROUPS OF TYPE ℝ3 ⋊ ℝ
Lee, Jong-Bum; Lee, Kyung-Bai; Shin, Joon-Kook; Yi, Seung-Hun;

Abstract
There are 7 types of 4-dimensional solvable Lie groups of the form $\small{{\mathbb{R}^3}\;{\times}_{\varphi}\;{\mathbb{R}}}$ which are unimodular and of type (R). They will have left. invariant Riemannian metrics with maximal symmetries. Among them, three nilpotent groups $\small{({\mathbb{R}^4},\;Nil^3\;{\times}\;{\mathbb{R}\;and\;Nil^4)}$ are well known to have lattices. All the compact forms modeled on the remaining four solvable groups $\small{Sol^3\;{\times}\;{\mathbb{R}},\;Sol_0^4,\;Sol_0^4\;and\;Sol_{\lambda}^4}$ are characterized: (1) $\small{Sol^3\;{\times}\;{\mathbb{R}}}$ has lattices. For each lattice, there are infra-solvmanifolds with holonomy groups 1, $\small{{\mathbb{Z}}_2\;or\;{\mathbb{Z}}_4}$. (2) Only some of $\small{Sol_{\lambda}^4}$, called $\small{Sol_{m,n}^4}$, have lattices with no non-trivial infra-solvmanifolds. (3) $\small{Sol_0^{4}}$ does not have a lattice nor a compact form. (4) $\small{Sol_0^4}$ does not have a lattice, but has infinitely many compact forms. Thus the first Bieberbach theorem fails on $\small{Sol_0^4}$. This is the lowest dimensional such example. None of these compact forms has non-trivial infra-solvmanifolds.
Keywords
Bieberbach Theorems;infra-homogeneous spaces;solvmanifold;
Language
English
Cited by
1.
LEFT-INVARIANT MINIMAL UNIT VECTOR FIELDS ON THE SEMI-DIRECT PRODUCT Rn,;

대한수학회보, 2010. vol.47. 5, pp.951-960
1.
LEFT-INVARIANT MINIMAL UNIT VECTOR FIELDS ON THE SEMI-DIRECT PRODUCT Rn, Bulletin of the Korean Mathematical Society, 2010, 47, 5, 951
2.
The bounding problem for infra-solvmanifolds, Topology and its Applications, 2016, 202, 397
3.
Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures, manuscripta mathematica, 2017
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