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- W4298157230 abstract "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 $varphi_{w}:mathfrak{g}^{r}rightarrowmathfrak{g}$ for every semisimple Lie algebra $mathfrak{g}$. Given two words $w_{1}inmathcal{L}_{r_{1}}$ and $w_{2}inmathcal{L}_{r_{2}}$, we define and study the convolution of the corresponding word maps $varphi_{w_{1}}*varphi_{w_{2}}:=varphi_{w_{1}}+varphi_{w_{2}}:mathfrak{g}^{r_{1}+r_{2}}rightarrowmathfrak{g}$. We show that for any word $winmathcal{L}_{r}$ 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." @default.
- W4298157230 created "2022-10-01" @default.
- W4298157230 creator A5049314441 @default.
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- W4298157230 date "2019-12-28" @default.
- W4298157230 modified "2023-10-17" @default.
- W4298157230 title "On singularity properties of word maps and applications to probabilistic Waring type problems" @default.
- W4298157230 doi "https://doi.org/10.48550/arxiv.1912.12556" @default.
- W4298157230 hasPublicationYear "2019" @default.
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