Finitely-presented 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演示。我们列出了该组的一组生成器，以及定义乘法的一组称为关联符的规则。虽然这提供了一个非常紧凑的陈述，但不幸的是，通常不可能回答关于该团体的某些问题，只给出了陈述。例如，没有算法来确定由有限表示给出的群是有限的还是无限的。对于这些问题是可判定的有限表示群的一类是双曲群。它们有一个几何定义，自然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的标准定义)任何随机群都是双曲群，随着表示的大小的增加，概率趋于1。在Gromov于20世纪80年代给出双曲群的定义之前，许多研究人员已经研究了小抵消表示，它是定义乘法的规则之间具有特别好的“重叠”性质的表示。我们将考察双曲群的各种决策问题，包括单词和共轭问题，试图理解如何利用小消去理论的技巧来证明给定的表示是双曲的，并找到各种决策问题的有效解决方案。我们还将研究开发随机演示的新模型，看看这些模型是否会产生比双曲群更广泛的群。

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.