The ‘degree of k-step nilpotence’ of a finite group G is the proportion of the tuples (x₁,…, x_k+1 ∈ G^k+1 for which the simple commutator [x₁, …, x_k+1] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Følner sequence if G is amenable. In our first main result we show that if G is finitely generated, then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent (Theorem 1.5). This generalises both an earlier result of the second author treating the case k = 1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if ϕ is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of ϕ is sparse in a certain sense (Theorem 5.1). In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below, then G is virtually k-step nilpotent (Theorem 1.19), answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients (Theorem 1.21), generalising a result of Gallagher.

在“的程度ķ -step幂零”有限群G ^是元组（比例X ₁，...，X _{ķ 1} ∈ ģ ^{ķ 1}的量，简单换向器[ X ₁，...，X _{ķ 1} ] 等于身份。在本文中，我们研究了无限群G的这个版本，其幂零度通过以各种自然方式对G进行采样来定义，例如随机游走，或者如果G是Følner 序列在我们的第一个主要结果中，我们证明如果G是有限生成的，则当且仅当G几乎是k步幂零时，k步幂零度为正（定理 1.5）。这概括了第二作者处理k = 1情况的早期结果和 Shalev 对有限群的结果，并使用了来自这两个早期结果的技术。我们还使用由 Leibman 和其他人开发的群的多项式映射的概念表明，幂零度的程度在很大程度上不依赖于采样方法（定理 1.12）。作为我们论证的一部分，我们通过证明如果ϕ是映射到无扭幂零群的多项式，则ϕ的根集在某种意义上是稀疏的（定理 5.1）。在我们的第二个主要结果中，我们考虑G是残差有限但不一定是有限生成的情况。这里我们证明，如果G的有限商的k步幂零的度数从下方一致有界，那么G实际上是k步幂零（定理 1.19），回答了 Shalev 的问题。作为我们证明的一部分，我们证明有限群的幂零度是商的次乘法（定理 1.21），概括了 Gallagher 的结果。