Random Generation: from Groups to Algebras |
Dr. Damien Sercombe |
Freitag, 17. Januar 2025 10:30Uhr |
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There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, with emphasis on finite simple groups. In this talk, based on joint work with Aner Shalev, we study similar notions for finite and profinite associative algebras. Let $k$ be a finite field. Let $A$ be a finite associative algebra over $k$, and let $P(A)$ be the probability that two random elements of $A$ will generate it. It is known that, if $A$ is simple, then $P(A) \to 1$ as $|A| \to \infty$. We extend this result for larger classes of finite associative algebras. For $A$ simple, we estimate the growth rate of $P(A)$ and find the best possible lower bound for it. We also study the random generation of $A$ by two special elements. Next, let $A$ be a profinite algebra over $k$. We show that $A$ is positively finitely generated if and only if $A$ has polynomial maximal subalgebra growth. Related quantitative results are also obtained. |