Mapping class groups and related structures

曲面是所有科学学科中的基本对象，包括物理、化学和生物学。在非数学的背景下，我们通常认为一个表面是某种边界，就像我们所说的地球表面或海洋表面。在这样的背景下，表面是位于三维空间中的固有的二维物体。对数学家来说，表面是在居民看来大致是平面的任何空间，在相同的意义上，早期人类认为地球是平的。曲面的示例包括球体和圆环(甜甜圈的曲面)。曲面的数学概念抓住了2维空间的基本概念，而不坚持与周围空间的任何参考。加深对物体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对象的自同构群中，即对象到其自身的所有函数或映射的集合，这些函数或映射保持了对象的本质特征。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)也出现在数学的许多不同领域的各种上下文中，特别是在代数和拓扑学中。我们描述一个自然而重要的例子。虽然表面本质上是2维的，但表面是所有3维空间(甚至是一类重要的4维空间)的构建块。通过所谓的Heegaard分裂构建3维空间依赖于能够通过从一个表面到其自身的相同副本的映射以一种良好的方式将两个表面粘合在一起。这些地图恰恰是S(Mod)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)也在数学中，特别是在群论中发挥着重要的作用，因为Mod(S)与其他重要的群类之间有许多深刻的相似之处。这些包括算术群(例如，矩阵群SL(n，Z)，它是向量空间的自同构群)和Aut(F)，自由群F的自同构群，它是代数中的基本对象。组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理论界正在进行的一个主题，其主要目的是通过回答一个重要的子群：超椭圆Torelli群SI(S)的关键问题来拓宽和加深对MoD(S)的结构的认识。当研究曲面上曲线的两个基本性质在曲面的映射下如何改变时，就出现了这一组。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)自然出现在经典辫子理论中，也出现在代数几何中。提案的这一部分的成功(将与Margalit联合)将在代数和几何/拓扑学之间提供强有力的联系。提案的第二个目标是以Aut(F)的子群为中心，表示为PIA(F)。Aut(F)中的Pia(F)子群是Mod(S)中SI(S)的恰当类比。这种类比似乎没有被探究过。因此，分析PIA(F)的结构是非常及时的。提案这一部分的成功将是对进一步深化MOD(S)和AUT(F)之间的类比的重大贡献。PI建议使用组合和几何群论的方法来实现这些目标。特别是，PI将研究SI(S)在由具有一定对称性的曲面上的曲线构造的空间上的作用。国际和平研究所已做好准备，处理正在审议的问题，并将在最近与马加利特和奇尔德斯的联合工作的基础上再接再厉。

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.