Asymptotic and algorithmic methods in group theory

该提案描述了与群论的算法和渐近方面有关的几个项目： * 构造具有“超越”性质的群（特别是群的无穷挠群）。 * 剩余有限群的复杂性。 * 最大增长群作用的渐近性和伯恩赛德性质 * 群的渐近锥。 * 顺从群的渐近性质。这些主题是密切相关的。例如，Higman嵌入和S-机将被用来构造非线性表示的扭群和研究群中文字问题的复杂性。现代群论是一个非常快速发展的数学领域，它积累了几何学、组合学、逻辑学和计算机科学的方法。我们的提案描述了需要所有这些领域的方法和想法的项目。特别是，构造一个非对称的无限扭群（这是群论中的一个突出问题）需要构造一种合适的图灵机（称为S-机），并采用几何群论的方法来找到模拟这些机器的群的合适的等价物。另一个来自几何学的想法（主要是由于格罗莫夫）是群的渐近锥的想法。我们将通过研究渐近锥来发现群的代数和算法性质。

The proposal describes several projects related to algorithmic and asymptotic aspects of group theory: * Constructing finitely presented groups with "transcendental" properties (in particular, finitely presented infinite torsion groups). * Complexity of residually finite groups. * Asymptotic and Burnside properties of group actions of maximal growth * Asymptotic cones of finitely presented groups. * Asymptotic properties of amenable groups.These topics are intimately related. For example, Higman embeddings and S-machines are going to be used to construct finitely presented torsion groups and in the study of complexity of the word problem in groups.Modern group theory is a very fast developing area of mathematics that accumulates methods from geometry, combinatorics, logic and computer science. Our proposal describes projects that are require methods and ideas from all these areas. In particular, constructing a finitely presented infinite torsion group (which is one of the outstanding problems in group theory) will require constructing a suitable kind of Turing machine (called S-machines) and adapting methods from geometric group theory to find suitable quotients of groups simulating these machines. Another idea coming from geometry (and due mostly to Gromov) is the idea of an asymptotic cone of a group. We are going to study asymptotic cones to discover algebraic and algorithmic properties of groups.