Combinatorial, geometric and probabilistic properties of groups

群生成群的概念本质上是一个代数群。然而，在某些情况下，从几何的角度来看这样一个群是富有成效的，就像在几何群论领域中经常发生的那样，或者从组合的角度来看，就像在算术组合学领域中一样。研究群上的概率过程也可以是问题的丰富来源，并导致离散概率。最近，这些不同观点之间出现了一些令人兴奋的联系，导致了一些突破和一些美丽的结果。本研究的主要目的是发展、增加和进一步利用这些联系。从几何学上观察一个由群生成的群的经典方法是将它看作一个图。如果G是具有有限对称生成集S的群，则Cayley图C（G，S）是其顶点集是G的元素集的图，其中x和y通过边连通当且仅当存在s E S使得x = ys。格罗莫夫的一个著名定理表明，这个图的某种渐近几何性质（多项式增长）等价于群G上的一个相当强的代数条件（虚拟零值）。人们也可以使用凯莱图来定义群G上的各种概率过程，例如随机游动，它们有各种物理解释，例如在电网络的背景下。另一方面，G上的渗流是C（G，S）的每条边根据某种概率分布随机删除或保留，然后研究所得到的随机图。这可以解释为水通过多孔石头的流动，或者病毒的传播。一个群的多项式增长率与随机游动和随机化的行为密切相关。一个关于群的组合观点有多强大的特别例子是被称为近似子群的对象。这些是在某种意义上在群运算下“近似封闭”的群的子集。在过去的15年里，近似子群的研究取得了相当大的进展，这导致了许多显着的应用，例如数论，随机矩阵理论和理论计算机科学等领域。近似群也可以被认为是多项式增长的“局部”版本，实际上是Breuillard的开创性成果，绿色和陶关于近似群可以用来表明，多项式增长的一个给定的区域的一个群体是足以暗示虚拟niliraries。这一结果是由Tessera和Tointon开发的，使用各种群论性质的“局部”版本，并应用于详细描述群上随机游动的某些精细尺度行为，验证和推广了Benjamini和Kozma的两个长期存在的理论。Hutchcroft和Tointon也运用了这个和其他的“局部”群论概念（例如将阿贝尔群的商取为子集而不是子群的概念）来分析有限群上的渗流，验证Benjamini的一个著名猜想的大多数情况。这个项目将，除其他外，寻求发展这种对群论的“局部”观点，以证明类似方向上的进一步有限性和定量结果。它福尔斯代数之间;几何与拓扑;和逻辑与组合学EPSRC研究领域。

The notion of a finitely generated group is at heart an algebraic one. However, in certain situations it turns out to be fruitful to view such a group from a geometric perspective, as is often the case in the field of geometric group theory, or from a combinatorial perspective, as in the field of arithmetic combinatorics. Studying probabilistic processes on groups can also be a rich source of problems and results in discrete probability. Recently, several exciting links between these different perspectives have emerged, leading to a number of breakthroughs and some beautiful results. The overriding theme of this research is to develop, add to, and further exploit these links.A classical way of viewing a finitely generated group geometrically is to view it as a graph. If G is a group with a finite symmetric generating set S then the Cayley graph C(G,S) is the graph whose vertex set is the set of elements of G, with x and y connected by an edge if and only if there exists s E S such that x = ys. A famous theorem of Gromov shows that a certain asymptotic geometric property of this graph (polynomial growth) is equivalent to a rather strong algebraic condition on the group G (virtual nilpotence).One can also use Cayley graphs to define various probabilistic processes on a group G, such as random walks, which have various physical interpretations, for example in the context of electric networks. Percolation on G, on the other hand, is where each edge of C(G,S) is either deleted or retained at random according to some probability distribution, and then the resulting random graph is studied. This can be interpreted in the context of the flow of water through a porous stone, or the spread of a virus. The rate of polynomial growth of a group turns out to be intimately connected to the behaviours of both random walks and percolation.A particular example of how powerful the combinatorial perspective on groups can be is provided by objects called approximate subgroups. These are subsets of a group that are 'approximately closed' under the group operation in a certain sense. There has been considerable progress in the study of approximate subgroups over the last 15 years, and this has led to a number of remarkable applications, for example to fields as diverse as number theory, random matrix theory and theoretical computer science.Approximate groups can also be thought of as a 'local' version of polynomial growth, and indeed a seminal result of Breuillard, Green and Tao about approximate groups can be used to show that polynomial growth on a given region of a group is enough to imply virtual nilpotence. This result has been developed by Tessera and Tointon, using 'local' versions of various group-theoretic properties, and applied to describe in detail certain fine-scale behaviours of random walks on groups, verifying and generalising two long-standing conjectures of Benjamini and Kozma. Hutchcroft and Tointon have also deployed this and additional 'local' group-theoretic notions (such as the notion of taking a quotient of an abelian group by a subset, rather than a subgroup) to analyse percolation on finite groups, verifying most cases of a famous conjecture of Benjamini.This project will, amongst other things, seek to develop this 'local' perspective on group theory, in an effort to prove further finitary and quantitative results in a similar direction. It falls between the Algebra; Geometry & Topology; and Logic & Combinatorics EPSRC research areas.