Finitely-presented groups

有限呈现群

• 批准号: 2095926
• 金额: --
• 依托单位: University of St Andrews
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2095926
• 关键词: Finitely; presented; groups

One of the main ways to represent an infinite group using only a finite amount of data is via a presentation. We list a set of generators for the group, and a set of rules called relators that define the multiplication. Whilst this gives a very compact representation, unfortunately it is not possible in general to answer certain questions about the group, given only the presentation. For example, there is no algorithm to determine whether a group given by a finite presentation is finite or infinite.One class of finitely-presented groups for which these problems are decidable is the class of hyperbolic groups. These have a geometric definition, and naturally act on a hyperbolic metric space. One remarkable fact about hyperbolic groups is that (using many different standard definitions of a random group) any random group is hyperbolic with probability tending to 1 as the size of the presentation grows. Unfortunately, it is undecidable in general whether or not a given group is hyperbolic.Before the definition of hyperbolic groups was given by Gromov in the 1980s, many researchers had studied small cancellation presentations, which are presentations with especially nice "overlap" properties between the rules defining the multiplication.We will look at a variety of decision problems for hyperbolic groups, including the word and conjugacy problem, seeking to understand how techniques from small cancellation theory can be used to show that a given presentation is hyperbolic, and to find effective solutions to various decision problems. We will also look at developing new models of random presentations, to see whether these produce a wider class of groups than just the hyperbolic groups.

仅使用有限数量的数据来表示无限群的主要方法之一是通过表示。我们列出了一组群的生成元，以及一组定义乘法的称为关系子的规则。虽然这给出了一个非常紧凑的表示，不幸的是，它是不可能在一般情况下回答有关该集团的某些问题，只有介绍。例如，没有算法来确定一个有限表示的群是有限的还是无限的。其中一类有限表示的群是双曲群。它们有一个几何定义，并且自然地作用于双曲度量空间。关于双曲群的一个值得注意的事实是，（使用随机群的许多不同标准定义）任何随机群都是双曲的，随着表示的大小增加，概率趋于1。不幸的是，一个给定的群是否是双曲群，一般来说是不可判定的。在Gromov于20世纪80年代给出双曲群的定义之前，许多研究人员研究了小消去表示，这是定义乘法的规则之间具有特别好的“重叠”性质的表示。我们将研究双曲群的各种判定问题，包括字和共轭问题，寻求理解如何使用来自小抵消理论的技术来表明给定的表示是双曲线的，并找到各种决策问题的有效解决方案。我们还将研究发展随机表示的新模型，看看这些模型是否能产生比双曲群更广泛的群。