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

We study singularity properties of word maps on semisimple algebraic groups and Lie algebras, generalizing the work of Aizenbud-Avni in the case of the commutator map. 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 $\varphi_{w_{1}}*\varphi_{w_{2}}:=\varphi_{w_{1}}+\varphi_{w_{2}}:\mathfrak{g}^{r_{1}+r_{2}}\rightarrow\mathfrak{g}$. We show that for any word $w\in\mathcal{L}_{r}$ of degree $d$, and any simple Lie algebra $\mathfrak{g}$ with $\varphi_{w}(\mathfrak{g}^{r})\neq0$, one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking $O(d^{4})$ self-convolutions of $\varphi_{w}$. We deduce that a group word map of length $\ell$ becomes (FRS) at $(e,\ldots,e)\in G^{r}$ after $O(\ell^{4})$ self-convolutions, for any semisimple algebraic group $G$. We furthermore bound the dimensions of the jet schemes of the fibers of Lie algebra word maps, and the fibers of group word maps in the case where $G=\mathrm{SL}_{n}$. For the commutator word $\nu=[X,Y]$, we show that $\varphi_{\nu}^{*4}$ is (FRS) for any semisimple Lie algebra, 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, and provide applications to $p$-adic probabilistic Waring type problems.