球在群中：体积、结构与增长
Balls in groups: volume, structure and growth
AI总结：
本文在Breuillard-Green-Tao的Gromov定理有限版本中给出了精确界，证明了多项式增长群的结构性质，并应用于顶点传递图理论等多个领域。
摘要：
我们在Breuillard、Green和Tao关于多项式增长群的Gromov定理的有限版本中给出了精确的界。具体来说，我们证明对于每个非负整数d，存在c=c(d)>0，使得如果G是一个具有包含单位元的有限对称生成集S的群，并且对于某个正整数n，有|S^n|≤cn^{d+1}|S|，那么存在正规子群H≤Γ≤G，使得H⊆S^n，Γ/H是d-幂零的（即具有长度为d且因子循环的中心列），并且[G:Γ]≤g(d)，其中g(d)表示GL_d(ℤ)的有限子群的最大阶。关于幂零性和指数的界都是精确的；先前最好的界是幂零性的O(d)和指数的d的无效函数。事实上，我们将此作为G的结构更详细的精细尺度描述的一小部分。这些结果在顶点传递图理论的各个方面有广泛的应用：渗流理论、随机游走、有限群的结构、有限顶点传递图的缩放极限……我们在本文中获得了其中一些应用，并在配套论文中处理其他应用。其中一些应用归功于或与其他作者合作。
英文摘要：
We develop a fine-scale structural theory for balls of polynomial volume in finitely generated groups. Our main result shows that a polynomial-volume hypothesis on a single ball implies that every larger ball can, after quotienting by a certain canonical finite normal subgroup, be embedded in a lattice of a Lie group whose connected component is nilpotent with sharply bounded index and dimension. These Lie groups and lattices form chains of successive quotients that are essentially unique, giving a quantitative fine-scale analogue of Losert's strengthening of the Mal'cev completion. This describes the fine structure of the group with sharp algebraic and quantitative precision; for example, it leads to sharp bounds in Breuillard, Green and Tao's finitary version of Gromov's polynomial-growth theorem. These results have a wide range of applications in various aspects of the theory of vertex-transitive graphs, including for instance random walks, percolation, and scaling limits. We obtain some of these applications in the present paper, and treat others in companion papers. Some are due to or joint with other authors.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
