Comparison theorems and orbit counting in hyperbolic geometry
Comparison theorems and orbit counting in hyperbolic geometry
复制标题
双曲几何中的比较定理和轨道计数
DOI：
--
复制
发表时间：
1998
期刊：
影响因子：
--
通讯作者：
Richard Sharp
中科院分区：
文献类型：
--
作者：
M. Pollicott;Richard Sharp
在这篇文章中，我们解决一个有趣的问题，双曲几何。这是比较与双曲流形的基本群相关的不同量的问题（例如字长，万有覆盖中的位移等）。渐近地我们的方法涉及到的想法从“热力学”遍历理论和自动机相关联的强马尔可夫群的tnixture。0.引言在[5]坎农表明，基本群体的许多流形的负曲率（包括，例如，紧凑的流形）的生成函数的字长是一个合理的功能。为了解释这一性质的含义，我们回顾Milnor在他的基础论文[13]中研究了数量N（n）= Card{g fE r Igl = n} -的增长，在那里他通过与泛覆盖中体积的增长进行比较来获得估计。坎农用纯粹的组合方法证明了生成函数是有理的。特别地，这意味着我们可以找到常数yBi和Ci，以及正整数ki（i = 1，. . ., N），使得N（n）= #1_1 Cinkitn。坎农方法的关键一步是将grol 1 p与一个自动机联系起来。在这篇文章中，我们将用“加权”的概念来增强这一点。这使我们能够利用著名的热力学形式主义理论来证明一些新的结果，这些结果可以被视为上述结果的加权类似物。人们早就知道，这种方法必须与符号动力学，马尔可夫划分，有限类型的子移位（cf。[1]，[20]，[21]，特别是Bourdon的工作[4]），并通过扩展到统称为“热力学形式主义”的动力系统领域的整个工具。在本文中，我们将详细阐述这种联系，作为我们分析的一个组成部分。设TEln，对于n > 2，表示n维双曲空间（即唯一的n维单连通的完备黎曼流形，其所有截面都是截面的）。1991年数学学科分类。初级20 F32、22 E40、58 E40;次级11 F72、20 F10、58 F20。
In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a tnixture of ideas from both "thermodynamic" ergodic theory and the automaton associated to strongly Markov groups. 0. INTRODUCTION In [5] Cannon showed that for fundamental groups of many manifolds of negative curvature (including, for example, compact manifolds) the generating function for word length is a rational function. To explain the implications of this property, we recall that the growth of the quantity N(n) = Card{g fE r Igl = n} -was studied by Milnor in his fundamental paper [13], where he obtained estimates using comparison with the growth of volume in the universal cover. Cannon used purely combinatorial methods to show that the generating function is rational. In particular, this implies that we can find constants yBi and Ci, and positive integers ki (i = :1, . . ., N) such that N(n) = #1_1 Cinkitn. The key step in Cannon's approach was to associate to the grol1p an automaton. In this article we shall augment this with the notion of "weighting". This allows us to draw upon the well-known theory of Thermodynamic Formalism to prove a number of new results which can be viewed as weighted analogues of the above results. It has long been understood that this approach must havew close connections with symbolic dynamics, Markov partitions, subshifts of finite type (cf. [1], [20],[21] and, in particular, the work of Bourdon [4]) and by extension to the whole paraphernalia of the area of dynamical systems collectively called "Thermodynamic Formalism". In this article, we shall elaborate on this connection as an integral part of our analysis. Let TEln, for some n > 2, denote n-dimensional hyperbolic space (i.e. the unique n-dimensional simply connected, complete Riemannian manifold wi-th all sectional Received by the editors May 23, 1995. 1991 Mathematics Subject Classification. Primary 20F32, 22E40, 58E40; Secondary 11F72, 20F10, 58F20.