Automorphic word maps and the Amit–Ashurst conjecture

Abstract In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), 6, 599–609 by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230–246 using elementary techniques.