We prove a quantitative, finitary version of Trofimov’s result that a connected, locally finite vertex-transitive graph Γ of polynomial growth admits a quotient with finite fibres on which the action of Aut(Γ) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph Γ of large diameter admits a quotient with fibres of small diameter on which the action of Aut(Γ) is virtually abelian with vertex stabilisers of bounded size. We also show that Γ has moderate growth in the sense of Diaconis and Saloff-Coste, which is known to imply that the mixing and relaxation times of the lazy random walk on Γ are quadratic in the diameter. These results extend results of Breuillard and the second author for finite Cayley graphs of large diameter. Finally, given a connected, locally finite vertex-transitive graph Γ exhibiting polynomial growth at a single, sufficiently large scale, we describe its growth at subsequent scales, extending a result of Tao and an earlier result of our own for Cayley graphs. In forthcoming work we will give further applications.

我们证明了Trofimov结果的定量的最终版本，即多项式增长的连接的局部有限的顶点-传递图Γ允许包含有限纤维的商，在该纤维上Aut（Γ）的作用实际上与有限顶点稳定器是幂等的。我们还介绍了一些应用程序。我们表明，一个大直径的有限连通顶点传递图Γ接纳一个小直径纤维的商，在该商数上，Aut（Γ）的作用实际上是阿贝尔式的，具有一定大小的顶点稳定器。我们还表明，在Diaconis和Saloff-Coste的意义上，Γ具有适度的增长，这已知意味着Γ上的惰性随机游走的混合和弛豫时间直径是平方的。这些结果扩展了Breuillard和第二作者关于大直径有限Cayley图的结果。最后，给定一个在单个足够大的尺度上展示多项式增长的连接的局部有限顶点传递图Γ，我们描述其在随后尺度上的增长，扩展了Tao的结果和我们自己对Cayley图的更早结果。在即将进行的工作中，我们将提供进一步的应用程序。