Aner Shalev

Random Generation: from Groups to Algebras

In recent decades there has been considerable interest in questions of random generation of finite and profinite groups, with emphasis on finite simple groups. I will give some background, and then discuss joint work with Damian Sercombe, studying similar notions for finite and profinite algebras.

Let $A$ be a finite associative, unital algebra over a (finite) field $k$. Let $P(A)$ be the probability that two random elements of $A$ will generate $A$ as a unital $k$-algebra. It was proved by Neumann and Praeger that, if $A$ is simple, then $P(A) \to 1$ as $|A| \to \infty$. We extend this result for larger classes of 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.

Turning to profinite algebras $A$, we show that $A$ is positively finitely generated if and only if $A$ has polynomial maximal subalgebra growth, and obtain related quantitative results.

If time permits I will also discuss random generation of group rings (answering a question of Rohrle) and of certain non-associative algebras (related to a question of Petersson on Jordan algebras).