Bieberbach theorems for solvable Lie groups
Bieberbach theorems for solvable Lie groups
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可解李群的比伯巴赫定理
DOI：
10.4310/ajm.2001.v5.n3.a6
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发表时间：
2001
影响因子：
0.6
通讯作者：
F. Raymond
中科院分区：
文献类型：
--
作者：
K. Dekimpe;K. Lee;F. Raymond
对于所有a：€ G.利用左不变向量场定义的G上的线性联络，已知Aff（G）是G的保联络超同态群，见[KT]命题2.1。术语.本文用李群G的格来表示G的离散余紧子群。进一步，我们将说李群G的自同构a是幂单的当且仅当它的微分da（在相应的李代数（&）的自同构群中）是幂单的。类似地，我们将讨论一个作用在李群G上的元素。对于G = IR，Bieberbach证明了以下三个定理。见[W]或[C]：定理I“。设irCMx 0（n）是一个格.则T = TT DM是W的格，F在TT中有有限指标。定理2.设7 r，7 r 'CMxi 0（n）是格.则每一个同构0：TT -）·TT'都是R × GL（n）的一个元素的共轭。R）。定理3“。在每个环面Z\E下，只有1000个平坦流形被环面覆盖。
for all a: € G. With the linear connection on G defined by the left invariant vector fields, it is known that Aff(G) is the group of connection-preserving diffeomorphisms of G, see [KT] Proposition 2.1. TERMINOLOGY. In this paper we will use the term lattice of a Lie group G, to denote a discrete cocompact subgroup of G. Further, we will say that an automorphism a of a Lie group G is unipotent if and only if its differential da (in the automorphism group of the corresponding Lie algebra (&) is unipotent. Analogously we will speak of an element acting unipotently on a Lie group G. For G = IR, the following three theorems have been proven by Bieberbach. See [W] or [C]: THEOREM I'. Let ir C M x 0(n) be a lattice. Then T = TT D M is a lattice of W, and F has finite index in TT. THEOREM 2. Let 7r,7r' C M xi 0(n) be lattices. Then every isomorphism 0 : TT -)■ TT' is a conjugation by an element ofR x GL(n. R). THEOREM 3'. Under each torus Z\E; there are only finitely many flat manifolds which are covered by the torus.