Combinatorial and probabilistic aspects of groups

80年代初格罗莫夫的一个开创性定理指出，有限生成群有多项式增长的充要条件是它几乎是幂零的。这反过来又在群的概率中有很深的应用，例如Varopoulos对简单随机游动在其上是常返的群的刻画，以及Dumil-Copin，Goswami，Raoufi，Severo和Yadin最近的突破表明，如果群不是虚拟循环的，群上的临界渗流概率严格小于1。事实上，这两个结果都利用Trofimov的一个结果推广到点传递图，该结果表明多项式增长的点传递图在某种意义上可以被Cayley图逼近。2011年，Breuillard，Green和Tao对Gromov定理进行了有限改进，这是他们关于近似群的著名工作的结果。Tessera和Tointon随后对Trofimov的结果进行了类似的改进。Hutchcroft和Tointon利用这些结果得到了类似于Dumil-Copin等人关于有限点传递图的结果，实质上验证了2001年Benjamini的一个猜想。在即将公布的工作中，Tessera和Tointon进一步发展了这一理论，并用它证明了Varopoulos结果的定量和有限版本，其中一部分是由Benjamini和Kozma在2002年猜测的，Easo和Hutchcroft用它来证明Schramm关于渗流的臭名昭著的局部性猜想。Tessera和Tointon将这一理论带到了大多数数量界限都可以证明是尖锐的地步。然而，该理论仍有许多方面可以磨砺，这个项目的一个目标是磨砺其中的一些。例如，陶已经证明了一组多项式增长的局部增长程度可以增加和减少至多有限次，变化的次数以初始增长率为界。Tessera和Tointon得到了次数可以减少的尖锐界限，但他们关于次数可以增加的界限可能并不是最优的。再举一个例子，Breuillard和Tointon证明了一个直径多项式大的有限群有一个‘大’商，它有一个有界阶和指数的交换子群。Tessera和Tointon得到了秩上的最优界，但它们在指数上的界可能不是最优界。此外，关于该集团的额外结构性假设似乎很可能导致该集团在排名上有更强的界限。这项研究的方法论本质上是非常新颖的，因为这些问题涉及到对一种本身尚未完全发表的理论的改进。鉴于这一理论最近在揭示概率论中古老而困难的猜想方面取得了惊人的成功，其影响的潜力也是巨大的。该项目属于EPSRC代数研究领域。

A seminal theorem of Gromov from the early 80s states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. This has in turn had deep applications in probability on groups, such as Varopoulos's characterisation of groups on which the simple random walk is recurrent, and Duminil-Copin, Goswami, Raoufi, Severo and Yadin's recent breakthrough showing that the critical percolation probability on a group is strictly less than 1 provided that the group is not virtually cyclic. In fact, both of these results extend to vertex-transitive graphs using a result of Trofimov showing that a vertex-transitive graph of polynomial growth can be approximated by a Cayley graph in a certain sense. In 2011, Breuillard, Green and Tao gave a finitary refinement of Gromov's theorem as a consequence of their celebrated work on approximate groups. Tessera and Tointon subsequently obtained a similar refinement of Trofimov's result. Hutchcroft and Tointon have used these results to obtain an analogue of Duminil-Copin et al's result for finite vertex-transitive graphs, essentially verifying a conjecture of Benjamini from 2001. In forthcoming but already announced work, Tessera and Tointon develop the theory further and use it to prove a quantitative and finitary version of Varopoulos's result, part of which was conjectured by Benjamini and Kozma in 2002, and Easo and Hutchcroft use it to prove Schramm's notorious locality conjecture for percolation. Tessera and Tointon have brought this theory to the point where most of the quantitative bounds are provably sharp. However, there are a number of aspects of the theory that could still be sharpened, and one aim of this project is to sharpen up some of these. For instance, Tao has shown that the 'local' growth degree of a group of polynomial growth can both increase and decrease at most finitely many times, with the number of changes bounded in terms of the initial growth rate. Tessera and Tointon obtain a sharp bound on the number of times the degree can decrease, but their bound on the number of times it can increase may not be optimal. For another example, Breuillard and Tointon proved that a finite group with a polynomially large diameter has a 'large' quotient that has an abelian subgroup of bounded rank and index. Tessera and Tointon obtain the optimal bound on the rank, but their bound on the index may not be optimal. Moreover, it appears likely that additional structural assumptions about the group should lead to stronger bounds on the rank. The methodology of this research is by its very nature highly novel, since these questions involve improving upon a theory that is itself not yet fully published. Given the spectacular recent success of this theory in shedding light on old and difficult conjectures in probability theory, the potential for impact is also significant. This project falls within the EPSRC Algebra research area.