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UNIMODULAR GROUPS OF TYPE ℝ3 ⋊ ℝ

  • Lee, Jong-Bum (Department of Mathematics Sogang University) ;
  • Lee, Kyung-Bai (Department of Mathematics University of Oklahoma Norman) ;
  • Shin, Joon-Kook (Department of Mathematics Chungnam National University) ;
  • Yi, Seung-Hun (Sciences and Liberal Arts-Mathematics Division Youngdong University)
  • 발행 : 2007.09.30

초록

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

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참고문헌

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피인용 문헌

  1. LEFT-INVARIANT MINIMAL UNIT VECTOR FIELDS ON THE SEMI-DIRECT PRODUCT Rn vol.47, pp.5, 2010, https://doi.org/10.4134/BKMS.2010.47.5.951
  2. The bounding problem for infra-solvmanifolds vol.202, 2016, https://doi.org/10.1016/j.topol.2016.02.001
  3. Lattices in almost abelian Lie groups with locally conformal Kähler or symplectic structures 2017, https://doi.org/10.1007/s00229-017-0938-3