Tommaso Toti | Searcharxiv

arXiv2026

Engel probability in wreath products of $p$-groups

We give upper and lower bounds for the number of solutions of the equation $e_n(x,y) = g$ in the group $W_k=(C_p\wr C_{p^k})^2$, where $e_n(x,y)$ is the $n$-th Engel word and $g\in W_k$. We obtain several corollaries from this. First, we prove a stronger version of the Amit-Ashurst conjecture for Engel words in $W_k$. We also prove that Engel words are not probabilistic identities in profinite groups with arbitrarily large wreath product quotients $W_k$. To conclude, we construct closed subsets of $(C_p\wr\Z_p)^2$ with positive Haar measure, empty-interior, and which are the preimage of an Engel word map.

math.GR↗

arXiv2026

On probabilistic identities and coset identities in pro-$p$ groups

It is shown that a probabilistic identity on a $σ$-compact $K$-analytic group $G$, $K$ a non-archimedean local field, is a coset identity. As an application, one concludes that compact $K$-analytic groups and various pro-$p$ groups obtained from free constructions satisfy a probabilistic Tits alternative. By means of Lie-theoretic methods, we also study torsion probabilistic identities in virtually free pro-$p$ and compact $p$-adic analytic groups.

math.GR↗