The structure and density of k $k$ -product-free sets in the free semigroup and group

The free semigroup $F$ on a finite alphabet $A$ is the set of all finite words with letters from $A$ equipped with the operation of concatenation. A subset $S$ of $F$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$. We prove that a $k$-product-free subset of $F$ has upper Banach density at most $1/\rho \left(k\right)$, where $\rho \left(k\right)=\min \{\ell :\ell \nmid k-1\}$. We also determine the structure of the extremal $k$-product-free subsets for all $k\notin \{3,5,7,13\}$; a special case of this proves a conjecture of Leader, Letzter, Narayanan, and Walters. We further determine the structure of all strongly $k$-product-free sets with maximum density. Finally, we prove that $k$-product-free subsets of the free group have upper Banach density at most $1/\rho \left(k\right)$, which confirms a conjecture of Ortega, Rué, and Serra.