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Mapping class groups and related structures

映射类组和相关结构

基本信息

批准号：
EP/J019593/1
负责人：
Tara Brendle
金额：
$ 10.59万
依托单位：
University of Glasgow
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ019593%2F1
关键词：
Mapping class groups related structures

项目摘要

Surfaces are fundamental objects in all scientific disciplines, including physics, chemistry, and biology. In a non-mathematical context, we usually think of a surface as a boundary of some sort, as when we speak of the surface of the earth or the sea. In such a context, a surface is an inherently 2-dimensional object situated in 3-dimensional space.To a mathematician, a surface is any space that appears roughly planar to an inhabitant, in the same sense in which early man believed the earth to be flat. Examples of surfaces include the sphere and the torus (the surface of a doughnut). The mathematical notion of a surface captures the fundamental idea of 2-dimensional space, without insisting on any reference to a surrounding space. A key way to deepen understanding of an object is to study its symmetries. These symmetries are encoded in an object's automorphism group, that is, the set of all functions or maps of an object to itself which preserve its essential features. The mapping class group Mod(S) of the surface S is the automorphism group of a surface, including, e.g., rotations. The mapping class group Mod(S) also appears in a wide variety of contexts in many different areas of mathematics, particularly in algebra and topology. We describe one natural and important example. Though inherently 2-dimensional, surfaces are the building blocks for all 3-dimensional spaces (and even an important class of 4-dimensional spaces) The construction of 3-dimensional spaces via a so-called Heegaard splitting depends on being able to "glue" two surfaces together in a nice way via a map from one surface to an identical copy of itself. These maps are precisely the elements of Mod(S). Thus one reason to study Mod(S) is to better understand 3-dimensional spaces.The mapping class group Mod(S) also plays an important role in mathematics and particularly in group theory because of the many deep analogies between Mod(S) and other important classes of groups. These include arithmetic groups (e.g., the matrix group SL(n,Z), which is an automorphism group of a vector space) and Aut(F), the automorphism group of a free group F, a fundamental object in algebra. The group Aut(F) also appears frequently in geometric contexts. An ongoing theme in the international geometric group theory community is to compare and contrast the various properties of these three classes of groups.The primary objective of the proposal is to broaden and deepen knowledge of the structure of Mod(S) by answering key questions about an important subgroup: the hyperelliptic Torelli group SI(S). This group arises when one studies how two basic properties of curves on a surface are changed under a map of the surface. These two properties are (1) symmetries under rotation by 180 degrees, and (2) their homology classes (homology is an algebraic invariant of a geometric/topological object). The mapping class group SI(S) appears naturally in the classical theory of braids and also in algebraic geometry. Success in this part of the proposal (which will be joint with Margalit) will provide a strong link between algebra and geometry/topology.A second objective of the proposal is centred on a subgroup of Aut(F), denoted PIA(F). The subgroup PIA(F) in Aut(F) is the appropriate analogue of SI(S) in Mod(S). This analogy does not appear to have been explored. Thus an analysis of the structure of PIA(F) is extremely timely. Success in this part of the proposal will represent a significant contribution to the furthering of analogies between Mod(S) and Aut(F). The PI proposes to use methods of combinatorial and geometric group theory in order to achieve these objectives. In particular, the PI will study the action of SI(S) on spaces constructed using curves on the surface having a certain symmetry property. The PI is well poised to address the questions under consideration, and will build on recent joint work with Margalit and Childers.

表面是所有科学学科的基本对象，包括物理、化学和生物学。在非数学的背景下，我们通常认为表面是某种边界，就像我们说地球或海洋的表面一样。在这种情况下，表面本质上是位于三维空间中的二维物体。对数学家来说，表面是任何在人类看来大致是平面的空间，就像早期人类认为地球是平的一样。表面的例子包括球体和环面（甜甜圈的表面）。曲面的数学概念抓住了二维空间的基本概念，而不坚持任何对周围空间的参考。加深对一个物体的理解的一个关键方法是研究它的对称性。这些对称性被编码在一个对象的自同构群中，也就是说，一个对象到它自身的所有函数或映射的集合，这些函数或映射保持了它的本质特征。曲面S的映射类群Mod(S)是曲面的自同构群，包括，例如，旋转。映射类组Mod(S)也出现在许多不同数学领域的各种上下文中，特别是在代数和拓扑中。我们描述一个自然而重要的例子。虽然表面本质上是二维的，但它是所有三维空间（甚至是一类重要的四维空间）的基石。通过所谓的heegard分裂来构建三维空间，取决于能否通过从一个表面到另一个表面的映射，以一种很好的方式将两个表面“粘”在一起。这些地图正是Mod(S)的元素。因此，研究Mod(S)的一个原因是为了更好地理解三维空间。映射类群Mod(S)在数学中也扮演着重要的角色，特别是在群论中，因为Mod(S)和其他重要的群类之间有许多深刻的相似之处。这包括算术群（如矩阵群SL(n，Z)，它是向量空间的自同构群）和自由群F的自同构群Aut(F)，这是代数中的一个基本对象。群Aut(F)也经常出现在几何环境中。比较和对比这三类群的各种性质是国际几何群论界的一个持续的主题。该提案的主要目标是通过回答关于一个重要子群：超椭圆Torelli群SI(S)的关键问题来拓宽和深化对Mod(S)结构的认识。当人们研究曲面上曲线的两个基本性质如何在曲面的映射下改变时，就会出现这一组。这两个性质是(1)旋转180度下的对称性，(2)它们的同调类（同调是几何/拓扑对象的代数不变量）。映射类群SI(S)在经典辫理论和代数几何中自然出现。提案的这一部分（将与Margalit联合）的成功将为代数和几何/拓扑之间提供强有力的联系。该提案的第二个目标集中在Aut(F)的子组上，表示为PIA(F)。Aut(F)中的子群PIA(F)与Mod(S)中的SI(S)类似。这个类比似乎没有被探索过。因此，对PIA(F)的结构进行分析是非常及时的。提案这一部分的成功将对进一步推进Mod(S)和Aut(F)之间的类比做出重大贡献。为了达到这些目标，PI建议使用组合和几何群论的方法。特别是，PI将研究SI(S)对使用具有一定对称性的曲面上的曲线构造的空间的作用。PI已准备好处理正在审议的问题，并将以最近与马加利特和蔡尔德斯的联合工作为基础。