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On Singularity Properties of Word Maps and Applications to Probabilistic Waring Type Problems

About this Title

Itay Glazer and Yotam I. Hendel

Publication: Memoirs of the American Mathematical Society
Publication Year: 2024; Volume 299, Number 1497
ISBNs: 978-1-4704-7043-2 (print); 978-1-4704-7875-9 (online)
DOI: https://doi.org/10.1090/memo/1497
Published electronically: July 11, 2024

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Table of Contents

Chapters

• 1. Introduction
• 2. Preliminaries
• 3. Some properties of the algebraic convolution operation
• 4. Lie algebra word maps
• 5. Lie algebra word maps: Proof of the main theorems
• 6. The commutator map revisited
• 7. Local behavior of word maps and a lower bound on the log canonical threshold of their fibers
• 8. Number theoretic interpretation of the flatness, $\varepsilon$-jet flatness and (FRS) properties
• 9. Applications to the $p$-adic probabilistic Waring type problem

Abstract

We study singularity properties of word maps on semisimple Lie algebras, semisimple algebraic groups and matrix algebras and obtain various applications to random walks induced by word measures on compact $p$-adic groups.

Given a word $w$ in a free Lie algebra $\mathcal {L}_{r}$, it induces a word map $\varphi _{w}:\mathfrak {g}^{r}\rightarrow \mathfrak {g}$ for every semisimple Lie algebra $\mathfrak {g}$. Given two words $w_{1}\in \mathcal {L}_{r_{1}}$ and $w_{2}\in \mathcal {L}_{r_{2}}$, we define and study the convolution of the corresponding word maps $\varphi _{w_{1}}*\varphi _{w_{2}}≔\varphi _{w_{1}}+\varphi _{w_{2}}:\mathfrak {g}^{r_{1}+r_{2}}\rightarrow \mathfrak {g}$.

By introducing new degeneration techniques, we show that for any word $w\in \mathcal {L}_{r}$ of degree $d$, and any simple Lie algebra $\mathfrak {g}$ with $\varphi _{w}(\mathfrak {g}^{r})\neq 0$, one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking $O(d^{4})$ self-convolutions of $\varphi _{w}$. Similar results are obtained for matrix word maps. We deduce that a group word map of length $\ell$ becomes (FRS), locally around identity, after $O(\ell ^{4})$ self-convolutions, for every semisimple algebraic group $\underline {G}$. We furthermore provide uniform lower bounds on the log canonical threshold of the fibers of Lie algebra, matrix and group word maps. For the commutator word $w_{0}=[X,Y]$, we show that $\varphi _{w_{0}}^{*4}$ is (FRS) for any semisimple Lie algebra, improving a result of Aizenbud-Avni, and obtaining applications in representation growth of compact $p$-adic and arithmetic groups.

The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form $\mathbb {Z}/p^{k}\mathbb {Z}$. This allows us to relate them to properties of some natural families of random walks on finite and compact $p$-adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to $p$-adic probabilistic Waring type problems.