Product-free subsets of profinite groups

Details Product-free subsets of profinite groups Product-free subsets of profinite groups

2012-02-19

arxiv.org/abs/1202.4194v2

Mathematics

Group Theory

Scientific paper

Gowers in his paper on quasirandom groups studies a question of Babai and Sos asking whether there exists a constant $c > 0$ such that every finite group $G$ has a product-free subset of size at least $c|G|$. Answering the question negatively, he proves that for sufficiently large prime $p$, the group $\mathrm{PSL}_2(\mathbb{F}_p)$ has no product-free subset of size $cn^{8/9}$, where $n$ is the order of $\mathrm{PSL}_2(\mathbb{F}_p)$. We will consider the problem for compact groups and in particular for profinite groups $\SL_k(\ZZ_p)$ obtain lower and upper exponential bounds for the supremal measure of the product-free sets. The proof involves establishing a lower bound for the dimension of non-trivial representations of the finite groups $\SL_k(\Z{p^n})$.

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