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Artin and Dehn twist subgroups of the mapping class group.

机译：Artin和Dehn扭曲了映射类组的子组。

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This dissertation investigates two types of subgroups in the mapping class group of an orientable surface. The first type of subgroups are isomorphic images of Artin groups. The second type of subgroups is one which is generated by three Dehn twists along simple closed curves with small geometric intersections.;Let S be a compact orientable surface. The mapping class group, Mod(S), of S is the group of isotopy classes of orientation preserving homeomorphisms of S fixing the boundary pointwise. Mod(S) is a very rich and complex object. In this dissertation, we make progress toward understanding the structure of the above mentioned subgroups of Mod(S).;We tackle three problems. The first problem focuses on finding embeddings of Artin groups into Mod(S). The second problem involves finding Artin relations of every length in Mod(S). And the third problem deals with understanding subgroups of Mod( S) generated by three Dehn twists along curves with small geometric intersections.;While it is easy to find nontrivial homomorphisms of Artin groups into Mod(S), the question of whether such homomorphisms are injective is quite hard. In this dissertation, we find embeddings of the Artin groups A (Bn), A (H3), A (I2(n)), and most notably A (An-1) into Mod(S). Further, we prove that if a collection { a1, ···, an} of simple closed curves in S has curve graph (see definition 4.1.2) An-1 and Nepsilon is a closed regular neighborhood of ∪ni=1 ai, then the subgroup of Mod( Nepsilon) generated by the (left) Dehn twists Ti along ai is isomorphic to A (An-1) almost all the time.;In the second problem, we study Artin relations in the mapping class group. If l ≥ 2 is an integer, then a and b satisfy the Artin relation of length l if aba··· = bab···, where each side of the equality has l terms. We give explicit elements of Mod(S) satisfying Artin relations of every integer length l ≥ 2. By direct computations, we find elements x and y in Mod (S) satisfying Artin relations of every even length ≥ 8 and every odd length ≥ 3. Then using the theory of Artin groups, we give two methods for finding Artin relations in Mod(S ). The first yields Artin relations of every length ≥ 3, while the second provides Artin relations of every even length ≥ 6. In the last two cases, we also show that x and y generate the Artin group A (I2(l)), where l is the length of the Artin relation satisfied by x and y.;The third problem is concerned with understanding subgroups in Mod(S) generated by three Dehn twists along curves with small geometric intersections. Let a1, a2, and a3 be distinct isotopy classes of essential simple closed curves in an orientable surface S. Assume that i(aj, ak) ∈ {0, 1, 2} for all j, k. Denote by Ti the (left) Dehn twist along ai, and let G represent the subgroup of Mod(S) generated by T1, T2, and T 3. Set (x12, x13, x23) = (i(a1, a2), i(a1, a3), i(a2, a3)). We find explicit presentations for G when (x12, x13, x23) = (0, 0, 0), (1, 0, 0), (2, 0, 0), (1, 0, 1), and (1, 1, 1). For the triple (2, 1, 0), there are two cases to consider (see subsections 7.8.1 and 7.8.2). In both cases, we are not able to find an explicit presentation for G. Nevertheless, we prove that G is a subgroup of some Artin group A . Moreover, using the computer algebra software Magma, we show that G is finitely presented and is isomorphic to a subgroup of infinite index in A . Although we have obtained similar partial results for the triples (2, 2, 0), (2, 1, 0), (2, 1, 1), (2, 2, 0), and (2, 2, 2), we do not include them in this dissertation.;While the three problems discussed above are seemingly disconnected, they are in fact intimately related. They reflect a beautiful interplay between Artin groups and mapping class groups.
机译：本文研究了可定向曲面的映射类组中的两种子组。第一种亚组是Artin组的同构图像。第二类是由沿着简单的闭合曲线，具有小的几何交点的三个Dehn扭曲生成的子组。让S为紧凑的可定向曲面。 S的映射类组Mod（S）是保持S的同胚性的方向的同位素类的组，S的同胚点固定了边界。 Mod（S）是一个非常丰富和复杂的对象。本文在理解上述Mod（S）子群的结构方面取得了进展。我们解决了三个问题。第一个问题集中在找到Artin组嵌入Mod（S）中。第二个问题涉及找到Mod（S）中每个长度的Artin关系。第三个问题涉及理解由三个Dehn扭曲沿着具有小的几何交点的曲线生成的Mod（S）的子组。虽然很容易将Artin组的非同质同构性找到到Mod（S）中，但这种同构性是否为内射很难。在本文中，我们发现将Artin组A（Bn），A（H3），A（I2（n））和最特别是A（An-1）嵌入到Mod（S）中。此外，我们证明，如果S中简单闭合曲线的集合{a1，…·an}具有曲线图（请参见定义4.1.2）An-1和Nepsilon是∪ni= 1 ai的闭合正则邻域，那么由（左）Dehn沿Ti沿Ti扭转Ti生成的Mod（Nepsilon）子组几乎一直都与A（An-1）同构。在第二个问题中，我们研究了映射类组中的Artin关系。如果l≥2是整数，则如果aba··= bab···，则a和b满足长度为l的Artin关系，其中等式的两边都有l个项。我们给出满足每个整数长度l≥2的Artin关系的Mod（S）的显式元素。通过直接计算，我们发现满足偶数长度≥8和奇数长度≥3的Mod（S）中满足Artin关系的元素x和y然后，使用Artin组理论，给出了两种在Mod（S）中查找Artin关系的方法。第一个提供每个长度≥3的Artin关系，第二个提供每个均匀长度≥6的Artin关系。在后两种情况下，我们还表明x和y生成Artin组A（I2（l）），其中l是x和y满足的Artin关系的长度;第三个问题与理解由三个Dehn扭曲沿着具有小的几何交点的曲线生成的Mod（S）中的子组有关。假设a1，a2和a3是可定向曲面S中基本简单闭合曲线的不同同位素类别。假设对于所有j，k，i（aj，ak）∈{0，1，2}。用Ti表示沿着ai的（左）Dehn扭曲，并且让G表示由T1，T2和T 3生成的Mod（S）的子组。设（x12，x13，x23）=（i（a1，a2）， i（a1，a3），i（a2，a3））。当（x12，x13，x23）=（0，0，0），（1，0，0），（2，0，0），（1，0，1）和（1 ，1、1）。对于三元组（2、1、0），有两种情况要考虑（见7.8.1和7.8.2小节）。在这两种情况下，我们都无法找到G的明确表示。尽管如此，我们证明G是某些Artin组A的子组。此外，使用计算机代数软件Magma，我们证明G是有限表示的，并且与A中无限索引的子群同构。尽管我们对三元组（2、2、0），（2、1、0），（2、1、1），（2、2、0）和（2、2、2）获得了类似的部分结果，虽然我们在本文中未将它们包括在内。虽然上面讨论的三个问题看似不相关，但实际上它们是密切相关的。它们反映了Artin组和制图类组之间的优美相互作用。

著录项

  • 作者

    Mortada, Jamil.;

  • 作者单位

    The Florida State University.;

  • 授予单位 The Florida State University.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2011
  • 页码 134 p.
  • 总页数 134
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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