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

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 L_r, it induces a word map φ_w : g^r → g for every semisimple Lie algebra g. Given two words w₁ ∈ L_r1 and w₂ ∈ L_r2, we define and study the convolution of the corresponding word maps φ_w1 ∗ φ_w2 := φ_w1 + φ_w2 : g^r1+r2 → g. By introducing new degeneration techniques, we show that for any word w ∈ L_r of degree d, and any simple Lie algebra g with φ_w(g^r) ≠ 0, one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O(d⁴) self-convolutions of φ_w. Similar results are obtained for matrix word maps. We deduce that a group word map of length ℓ becomes (FRS), locally around identity, after O(ℓ⁴) self-convolutions, for every semisimple algebraic group 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 w₀ = [X,Y], we show that φ^∗_w⁴₀ 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 ℤ/p^kℤ. 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.