Rank-1 phenomena for mapping class groups
Rank-1 phenomena for mapping class groups
复制标题
映射类组的 Rank-1 现象
DOI：
10.1215/s0012-7094-01-10636-4
复制
发表时间：
2001
影响因子：
2.5
通讯作者：
Y. Minsky
中科院分区：
文献类型：
--
作者：
B. Farb;A. Lubotzky;Y. Minsky
设G是亏格g ≥ 1的闭可定向连通曲面。映射类群Mod（Bogg）是Bogg的保向同胚的合痕类的群Homeo（Bogg）/Homeo 0（Bogg）。将群Mod（Bg）及其在Teichmuller空间T（Bg）上的作用与单李群中的格及其在相关对称空间上的作用进行比较是一个反复出现的主题。事实上，群Mod（Bogg）共享半单李群中（算术）格的许多性质。例如，它们满足Tits替代，它们有有限的虚上同调维数，它们是剩余有限的，并且它们的每个可解子群是多环的。简单李群中的格之间的一个众所周知的二分法是一阶群中的格和更高阶格（即R阶至少为二的简单李群中的那些格）之间的二分法。这是有点神秘的Mod（Mod）是否类似于前者或后者。模空间的尖点结构、模具有塞尔性质（FA）[CV]的事实以及Ivanov版本（例如，见[Iv 2]）的关于高阶建筑物自同构群的Tits定理的Mod（Bogg）表示Mod（Bogg）的一些高阶行为。在这篇笔记中，我们增加了两个属性的列表（见，例如[Iv 1，Iv 2，Iv 3]和其中的参考文献）的性质，表现出相似性的Mod（Bogg）与格在秩一组：每个无限阶元素的Mod（Bogg）有线性增长的字度量，和Mod（Bogg）不受约束Bogg部分由NSF资助DMS 9704640和斯隆基金会奖学金支持。[2]部分由美国-以色列BSF赠款支持。部分由NSF资助DMS 9971596
Let Σg be a closed, orientable, connected surface of genus g ≥ 1. The mapping class group Mod(Σg) is the group Homeo(Σg)/Homeo0(Σg) of isotopy classes of orientation-preserving homeomorphisms of Σg. It has been a recurring theme to compare the group Mod(Σg) and its action on the Teichmuller space T (Σg) to lattices in simple Lie groups and their actions on the associated symmetric spaces. Indeed, the groups Mod(Σg) share many of the properties of (arithmetic) lattices in semisimple Lie groups. For example they satisfy the Tits alternative, they have finite virtual cohomological dimension, they are residually finite, and each of their solvable subgroups is polycyclic. A well-known dichotomy among the lattices in simple Lie groups is between lattices in rank one groups and higher-rank lattices, i.e. those lattices in simple Lie groups of R-rank at least two. It is somewhat mysterious whether Mod(Σg) is similar to the former or the latter. Some higher rank behavior of Mod(Σg) is indicated by the cusp structure of moduli space, by the fact that Mod(Σg) has Serre’s property (FA) [CV], and by Ivanov’s version (see, e.g. [Iv2]) for Mod(Σg) of Tits’s Theorem on automorphism groups of higher rank buildings. In this note we add two more properties to the list (see, e.g. [Iv1, Iv2, Iv3] and the references therein) of properties which exhibits similarities of Mod(Σg) with lattices in rank one groups: every infinite order element of Mod(Σg) has linear growth in the word metric, and Mod(Σg) is not bound∗Supported in part by NSF grant DMS 9704640 and by a Sloan Foundation fellowship. †Supported in part by the US-Israel BSF grant. ‡Supported in part by NSF grant DMS 9971596