Rank one lattices whose parabolic isometries have no rotational part
Rank one lattices whose parabolic isometries have no rotational part
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抛物线等轴无旋转部分的一阶晶格
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发表时间：
1998
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通讯作者：
Christoph Hummel
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作者：
Christoph Hummel
本文证明了一秩格的有限指数子群的一个结果，这个结果是由尖闭构造引起的。设X总是表示非紧型的秩1对称空间，即，X是双曲空间KH，n ≥ 2，其中K是R，C，H或O，在后一种情况下n = 2。我们总是用来表示等距群Iso（X）中的格，即X的等距群Iso（X）的离散子群，使得商X具有有限体积。下面的结果只与非均匀格有关，即其中X是非紧的格。对于每个X，这样的格都是由A的结果而存在的。波莱尔（见[B1]或[R]中的第十四章）。我们说抛物等距σ ∈ Iso（X）没有旋转部分，如果σ包含在某个Iwasawa分解Iso（X）= NAK的幂零部分N中.这里Iso（X）表示Iso（X）的单位分量。本文给出了一个简单的几何论证以证明下面的定理，它与Borel和加兰& Raghunathan的一个结果有关。定理对于非紧型秩一对称空间X的等距群中的任何格F < Iso（X），存在抛物等距的有限子集F，使得下列成立。设F ′ C是正规子群，且F ′ C = F。则在π ′中的任何抛物等距都没有转动部分。备注。定理的证明提供了一个确定F的显式过程，通过最后的例子，F在某种意义上是最优的。作为定理的一个推论，我们得到下列推论。该陈述在[GR]，引理6.5中得到证明，其中命题17.6来自[B2]，并且方法具有代数性质。推论（Borel and加兰& Raghunathan）.设V = X是一个完备的局部秩为1的非紧型有限体积对称流形，用N表示Iso（X）的Iwasawa分解的幂零部分.则存在一个有限正则覆盖V_n → V，使得V_n的每个尖点对某个格Γ < N都与ΓN × [0，∞）同构.编辑于1996年12月7日收到，修订版于1997年1月22日收到。1991年数学学科分类。小学53 C35;中学22 E40，22 E25。
We prove a result on certain finite index subgroups of rank one lattices which is motivated by cusp closing constructions. Let X always denote a rank one symmetric space of non-compact type, i.e., X is the hyperbolic space KH, n ≥ 2, where K is either R,C,H or O and n = 2 in the latter case. By Σ we always denote a lattice in the isometry group Iso(X), that is, a discrete subgroup of the isometry group Iso(X) of X such that the quotient ΣX has finite volume. The result below is only relevant for non-uniform lattices, i.e. lattices Σ where ΣX is non-compact. Such lattices exist for each X by a result of A. Borel (see [B1] or Chapter XIV in [R]). We say that a parabolic isometry σ ∈ Iso(X) has no rotational part if σ is contained in the nilpotent part N of some Iwasawa decomposition Iso(X) = NAK. Here Iso(X) denotes the identity component of Iso(X). In this note we give a simple geometric argument in order to prove the theorem below, which is related to a result of Borel and Garland & Raghunathan. Theorem. For any lattice Σ < Iso(X) in the isometry group of a rank one symmetric space X of non-compact type there exists a finite subset F ⊂ Σ of parabolic isometries such that the following holds. Assume Σ′ C Σ is a normal subgroup and Σ′ ∩ F = ∅. Then any parabolic isometry in Σ′ has no rotational part. Remark. The proof of the theorem provides an explicit procedure to determine F , which is in some sense optimal by the example at the end. As a consequence of the theorem we obtain the following corollary. The statement is proved in [GR], Lemma 6.5, with Proposition 17.6 from [B2], and the methods are of algebraic nature. Corollary (Borel and Garland & Raghunathan). Let V = ΣX be a complete, locally rank one symmetric manifold of non-compact type and finite volume, and denote by N the nilpotent part of an Iwasawa decomposition of Iso(X). Then there exists a finite regular covering V̂ → V such that each cusp of V̂ is diffeomorphic to ΓN × [0,∞) for some lattice Γ < N . Received by the editors December 7, 1996 and, in revised form, January 22, 1997. 1991 Mathematics Subject Classification. Primary 53C35; Secondary 22E40, 22E25.