Combinatorial, geometric and probabilistic properties of groups

从本质上讲，有限生成群的概念是一个代数概念。然而，在某些情况下，事实证明，从几何角度来看待这样一个群是卓有成效的，就像几何群论领域中经常出现的情况一样，或者从组合角度来看，例如在算术组合学领域中。研究群上的概率过程也可能是问题的丰富来源，并导致离散概率。最近，这些不同视角之间出现了几个令人兴奋的联系，导致了一些突破和一些美丽的结果。这项研究的首要主题是发展、添加和进一步利用这些联系。从几何角度观察有限生成的群的经典方法是将其视为图形。如果G是具有有限对称生成集S的群，则Cayley图C(G，S)是这样的图，其顶点集是G的元素集，其中x和y由一条边连接当且仅当存在S E S使得x=ys。Gromov的一个著名定理表明，该图的某种渐近几何性质(多项式增长)等价于群G(虚幂零性)上的一个相当强的代数条件。人们还可以使用Cayley图来定义群G上的各种概率过程，例如随机行走，这些过程具有不同的物理解释，例如在电网络的背景下。另一方面，G上的渗流是指C(G，S)的每条边按照某种概率分布被随机删除或保留，然后研究所得到的随机图。这可以在水通过多孔石的流动或病毒传播的背景下得到解释。一个群的多项式增长率被证明与随机游动和渗流的行为密切相关。一个关于群的组合观点可以多么强大的具体例子是由被称为近似子群的对象提供的。在某种意义上，这些是在群运算下被“近似封闭”的群的子集。在过去的15年里，对近似子群的研究已经取得了相当大的进展，这导致了许多显著的应用，例如在诸如数论、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，事实上，Breuillard，Green和Tao关于近似群的一个开创性结果可以用来证明群的给定区域上的多项式增长足以隐含虚拟幂零。这一结果是由Tessera和Tointon发展的，使用了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行为，验证和推广了Benjamini和Kozma的两个长期存在的猜想。Hutchcroft和Tointon也运用了这个概念和其他的‘局部’群论概念(例如用一个阿贝尔群的一个子集而不是一个子群取商的概念)来分析有限群上的渗流，验证了本杰明的一个著名猜想的大多数情况。这个项目将寻求发展这个关于群论的‘局部’观点，努力在类似的方向上进一步证明有限的和定量的结果。它介于代数、几何与拓扑学和逻辑与组合学EPSRC研究领域之间。

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.