The smallest arithmetic hyperbolic three-orbifold
The smallest arithmetic hyperbolic three-orbifold
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最小算术双曲三环
DOI：
10.1007/bf01389265
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发表时间：
1986
影响因子：
3.1
通讯作者：
E. Friedman
中科院分区：
文献类型：
--
作者：
T. Chinburg;E. Friedman
在本文中，我们确定了最小体积的完整的、可定向的、算术双曲3-轨道M 0 。下面将给出M o 的精确描述。事实上，我们将证明，M o 的体积比任何算术轨道折叠都要小，该算术轨道折叠被构造为一些上半平面和半空间的乘积的不可约保因子商；这些商将在下面精确定义。我们的证明完全是数论的，并且依赖于此类轨道折叠体积的 Borel [1] 公式。在后面的论文中，我们将应用相同的技术来生成前几个最小的完全可定向算术双曲 3 流形的列表。分别将 V、VO、VA 和 VAO 定义为所有完整、可定向、有限体积双曲 3 流形、3 轨道、算术 3 流形和算术 3 轨道的体积集。根据 Borel 定理 [1]，VA 和 VAO 是 R 的离散子集。根据 Jorgenson 和 Thurston 定理 [10, 14]，V 和 VO 是 F 的闭良序子集。订单类型 co'。因此，特别地，V、VO、VA和VAO中的每一个都存在最小的元素，并且V和VO的大多数元素不在VA或VAO中。尽管如此，M o 的体积（我们将证明它是 VAO 的最小元素）也是我们在撰写本文时所知道的 VO 的最小元素。此外，截至撰写本文时我们已知的 V 的最小和第二小元素都位于 VA 中（参见[9,18,2]）。了解 V 或 VO 的最小元素是否分别在 VA 或 VAO 中会很有趣。在这个方向上，Meyerhoff [10] 最近证明了当 ~ 3 是单位的本原立方根时，最小尖瓣完全可定向双曲 3 轨道折叠是基本群 PGL2 (Z [~ 3]) 的算术轨道折叠。我们现在精确定义本文要考虑的轨道折叠。设a和b为非负整数，且a+b>1。设H，b=(H2)ax(H3)b，其中H 2 和H 3 分别表示双曲上半平面和双曲上半空间。群 G~, b= PGL2 (~)" x PGL2 (IE) b 是 H,, b 的等距群，它保留了每个因子和三个-的方向
In this paper we determine the complete, orientable, arithmetic hyperbolic 3-orbifold M 0 of minimal volume. A precise description of M o will be given below. We will show, in fact, that M o has smaller volume than any arithmetic orbifold constructed as an irreducible factor-preserving quotient of the product of some number of upper half planes and half spaces; these quotients will be defined precisely below. Our proof is entirely number theoretic, and relies on a formula of Borel [1] for the volumes of such orbifolds. In a later paper, we will apply the same techniques to produce a list of the first few smallest complete orientable arithmetic hyperbolic 3-manifolds. Define V, VO, VA and VAO to be the set of volumes of all complete, orientable, finite-volume hyperbolic 3-manifolds, 3-orbifolds, arithmetic 3-manifolds and arithmetic 3-orbifolds, respectively. By a Theorem of Borel [1], VA and VAO are discrete subsets of R. By a Theorem of Jorgenson and Thurston [10, 14], V and VO are closed well-ordered subsets of F,. of order type co'. Thus, in particular, there is a smallest element of each of V, VO, VA and VAO, and most elements of V and VO are not in VA or VAO. Nonetheless, the volume of M o, which we will prove to be the smallest element of VAO, is the smallest element of VO that we know of as of this writing. Further, both the smallest and second smallest elements of V that are known to us as of this writing lie in VA (cf.[9, 18, 2]). It would be interesting to know whether or not the smallest element of V or VO is in VA or VAO, respectively. In this direction, Meyerhoff [10] has recently shown the smallest cusped complete orientable hyperbolic 3-orbifold to be the arithmetic orbifold with fundamental group PGL2 (Z [~ 3]) when~ 3 is a primitive cube root of unity. We now define precisely the orbifolds to be considered in this paper. Let a and b be non-negative integers with a+ b> 1. Let H,, b=(H2) ax (H3) b, where H 2 and H 3 denote the hyperbolic upper half plane and the hyperbolic upper half space, respectively. The group G~, b= PGL2 (~)" x PGL2 (IE) b is the group of isometries of H,, b which preserve each factor and the orientation of the three-