Growth of Unbounded Subsets in Nilpotent Groups, Random Mapping Statistics and Geometry of Group Laws

Abstract

Abstract For a group $G$ let $\gamma ^{\max }_G(n)$ denote the maximum number of length-$n$ words over an arbitrary $n$-letter subset of $G$. If $G$ is finitely generated and residually finite, then either $\gamma ^{\max }_G(n)$ is exponentially bounded, if and only if $G$ is virtually abelian, or $\gamma ^{\max }_G(n)\geq ~\left (e^{-\frac {1}{4}}+o(1)\right )n^n$; the latter bound cannot be improved, namely, it is attained for the Heisenberg group. For higher-step free nilpotent groups we have $(1-o(1))n^n\leq \gamma ^{\max }_G(n)\lneq n^n$. As a key ingredient in the proof, we calculate the number of pair histograms of functions $f\colon [n]\rightarrow [n]$ and the probability that a random function $f\colon [n]\rightarrow [n]$ can be uniquely determined by its pair histogram. We analyze group laws of the Heisenberg group by means of closed paths attached to words in the free group and their projected oriented polygons.