On singularity properties of word maps and applications to probabilistic Waring type problems (Q6605404)

Let \(F_{r}\) be a free group, \(\mathcal{L}_{r}\) a free Lie algebra, \(\mathcal{A}_{r}\) a free associative algebra, all of rank \(r\) and let \(w\) be a word, that is an element of \(F_{r}\), \(\mathcal{L}_{r}\), or \(\mathcal{A}_{r}\). The word \(w\) induces a word map \(\varphi_{w}\) when applied to a group, a Lie algebra or an associative algebra, respectively.\N\NThe paper is dedicated to the study of word maps on semisimple Lie algebras, semisimple algebraic groups, matrix algebras and compact \(p\)-adic groups. In particular, the authors focus on the singularity properties of such word maps and of the maps \(\varphi_{w^{\ast t}}\) which are obtained by successive self-convolutions \(w^{\ast t}=w\ast w\ast \ldots \ast w\) of the word \(w\). Here, convolution just means concatenation (resp. summation) of group (resp. algebra) words in distinct variables.\N\NIn [\textit{A. Aizenbud} and \textit{N. Avni}, Invent. Math. 204, No. 1, 245--316 (2016; Zbl 1401.14057); correction ibid. 227, No. 3, 1431-1434 (2022; Zbl 1482.14011)], the commutator word \(w=[X,Y]\) was investigated and it was shown that after \(21\) self-convolutions, the word map \(\varphi_{w^{\ast 21}}\) has mild singularities (it is flat, with fibers of rational singularities) when applied to any semisimple algebraic group or Lie algebra.\N\NIn this paper the authors generalize the result cited above to arbitrary word maps by introducing new degeneration techniques, as well as a systematic way to execute them. Using their previous results (see [Sel. Math., New Ser. 25, No. 1, Paper No. 15, 41 p. (2019; Zbl 1473.14027); J. Lond. Math. Soc., II. Ser. 103, No. 4, 1453--1479 (2021; Zbl 1482.14001)]) to deal with the situations where the word maps are wild. Using these methods, they were also able to give a simple alternative proof of the commutator result, with improved bounds.\N\NThe reviewer reports here a representative result in the case of Lie algebras. Theorem A: Let \(\{w_{i}\in \mathcal{L}_{r_{i}} \}_{i \in \mathbb{N}}\) denote a collection of Lie algebra words of degree at most \(d\). Then there exists \(0 < C < 10^{6}\), such that for any simple \(K\)-Lie algebra \(\mathfrak{g}\) for which \(\{ \varphi_{w_{i}} : \mathfrak{g}_{r_{i}} \rightarrow \mathfrak{g} \}\) are non-trivial, we have the following: (1) if \(m \geq Cd^{3}\), then \(\varphi_{w_{1}}\ast \ldots \ast \varphi_{w_{m}}\) is a flat morphism; (2) if \(m \geq Cd^{4}\), then \(\varphi_{w_{1}}\ast \ldots \ast \varphi_{w_{m}}\) is a flat morphism with reduced fibers of rational singularities (see Definition 1.5).\N\NIn the setting of compact \(p\)-adic groups, the authors study the family of random walks induced by word measures and obtain various uniform bounds on \(L^{\infty}\) and \(L^{1}\) mixing times, as well as uniform upper bounds on the probability densities.