## Probabilistic Waring problems for finite simple groups.(English)Zbl 1448.20063

Summary: The probabilistic Waring problem for finite simple groups asks whether every word of the form $$w_1w_2$$, where $$w_1$$ and $$w_2$$ are non-trivial words in disjoint sets of variables, induces almost uniform distributions on finite simple groups with respect to the $$L^1$$ norm. Our first main result provides a positive solution to this problem.
We also provide a geometric characterization of words inducing almost uniform distributions on finite simple groups of Lie type of bounded rank, and study related random walks.
Our second main result concerns the probabilistic $$L^\infty$$ Waring problem for finite simple groups. We show that for every $$l\geq 1$$, there exists (an explicit) $$N=N(l)=O(l^4)$$, such that if $$w_1,\dots,w_N$$ are non-trivial words of length at most $$l$$ in pairwise disjoint sets of variables, then their product $$w_1\cdots w_N$$ is almost uniform on finite simple groups with respect to the $$L^\infty$$ norm. The dependence of $$N$$ on $$l$$ is genuine. This result implies that, for every word $$w=w_1\cdots w_N$$ as above, the word map induced by $$w$$ on a semisimple algebraic group over an arbitrary field is a flat morphism.
Applications to representation varieties, subgroup growth, and random generation are also presented. In particular, we show that, for certain one-relator groups $$\Gamma$$, a random homomorphism from $$\Gamma$$ to a finite simple group $$G$$ is surjective with probability tending to $$1$$ as $$|G| \to \infty$$.

### MSC:

 20P05 Probabilistic methods in group theory 20D06 Simple groups: alternating groups and groups of Lie type 20C33 Representations of finite groups of Lie type 20G40 Linear algebraic groups over finite fields
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### References:

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