Product Decompositions of Groups

该项目的主要方面是考虑将有限和无限群的族分解为循环、阿贝尔或可解子群的乘积。历史背景来自伊藤的定理，拉扎德的工作分析pro-p群和有界生成之间的联系和同余子群性质的算术群建立的Lubotzky，Platonov和Rapinchuk。虽然PI认为这些主要结果的动机问题，他还调查循环，阿贝尔和可解分解在更广泛的背景下：有限简单群，无限置换群，几何群和团体的树木。除了相当令人惊讶的结构（例如，无限对称群是许多阿贝尔子群的乘积），这样的分解的存在导致强渐近限制（例如子群增长）。另一方面，试图表明某种分解不存在迫使人们从一个新的角度来看待子群结构，然后可以进一步分析。一个例子是Hausdorff维数的群体作用于根树。任何物体或结构的对称性集合形成一个群。因此，当人们需要分析对称性时（例如在量子物理和化学中，或者在几何、数论、编码理论和拓扑学中的数学中），群论自然就会出现。该项目的主要目标是在组族中找到某些“坐标系”，并使用简单的结构以透明的方式生成这些组，或者表明这种好的系统不存在。一个基本的例子是，每一个平面等距可以作为最多三个反射的合成而不是作为两个反射的合成来获得。在这种“坐标系”的存在和各种其他有趣的群性质之间存在着多种联系。除了深入和有用的结果，该项目也有基本的方面，可以在本科水平，从而服务于教育目的。

The main aspect of the project is to consider decomposing families of finite and infinite groups as the product of cyclic, Abelian or solvable subgroups. The historical background comes from Ito's theorem, Lazard's work on analytic pro-p groups and the connection between bounded generation and the congruence subgroup property for arithmetic groups established by Lubotzky, Platonov and Rapinchuk. While the PI considers problems motivated by these major results, he also investigates cyclic, Abelian and solvable decompositions in a broader context: for finite simple groups, infinite permutation groups, geometric groups and groups acting on trees. Besides rather surprising constructions (e.g., the infinite symmetric group is the product of finitely many Abelian subgroups), the existence of such a decomposition leads to strong asymptotic restrictions (e.g. on subgroup growth). On the other hand, trying to show that a certain decomposition does not exist forces one to look at the subgroup structure from a new point of view which then can be analyzed further. An example for this is Hausdorff dimension of groups acting on rooted trees. The set of symmetries of any object or structure forms a group. Thus group theory naturally comes into the picture whenever one needs to analyze symmetries (e.g. in quantum physics and chemistry or inside mathematics in geometry, number theory, coding theory and topology). The main goal of the proposed project is to find certain 'coordinate systems' in families of groups and so generate these groups in a transparent way using simple structures or to show that such nice systems do not exist. An elementary example is the fact that every plane isometry can be obtained as the composition of at most three reflections but not as the composition of two. There are manifold connections between the existence of such 'coordinate systems' and various other interesting group properties. Other than deep and useful results, the project also has elementary aspects which can be presented on an undergraduate level and thus serve educational purposes.