Counting commensurability classes of hyperbolic manifolds
Counting commensurability classes of hyperbolic manifolds
复制标题
计算双曲流形的可公度类
DOI：
10.1007/s00039-014-0294-3
复制
发表时间：
2014
影响因子：
2.2
通讯作者：
Arie Levit
中科院分区：
文献类型：
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作者：
T. Gelander;Arie Levit
格罗莫夫和皮亚特斯基-夏皮罗证明了任何给定维度的有限体积非算术双曲流形的存在。在四维及更高维度中，我们证明大约存在 vv 个这样的体积最多为 v 的流形，考虑到可通约性。由于算术流形的数量往往是多项式，因此几乎所有双曲流形在适当的意义上都是非算术流形。此外，通过限制对非紧流形的关注，我们的结果意味着 SO(n, 1) 中晶格的准等距类数量具有相同的增长类型。我们的方法涉及几何空间图的构造，该构造依赖于某些二次形式的算术属性。
Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about vv such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasi-isometry classes of lattices in SO(n, 1). Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.
DOI：
10.4007/annals.2010.172.2197
发表时间：
2008-11
期刊：
arXiv: Group Theory
影响因子：
--
作者：
M. Belolipetsky;T. Gelander;A. Lubotzky;A. Shalev
通讯作者：
M. Belolipetsky;T. Gelander;A. Lubotzky;A. Shalev