Random free semigroups of affine groups

Abstract

We investigate random freeness of semigroups in the solvable non-virtually nilpotent setting. We focus on a correlated affine model, namely semigroups generated by the two components of a finitely supported random walk $L_n=(L_{n,1},L_{n,2})$ on $\operatorname{Aff}(K)^2$ whose two components share a common linear part. In this model, we show that the long-term behavior of freeness is completely governed by an abelian shadow, namely the projected random walk on the multiplicative group $A\subset K^\times$ generated by the common multipliers. If this walk is transient, then {the semigroup} $\langle L_{n,1},L_{n,2}\rangle_+$ is eventually free almost surely. If it is recurrent, then freeness does not stabilize: the set of non-free times is almost surely infinite, yet has almost sure density zero. Moreover, the obstructions to freeness depends solely on the common linear part and admits an explicit arithmetic description in terms of roots of Littlewood polynomials. The proof combines a local contraction theorem for affine random walks over arbitrary local fields, developed in the appendix and related to the theory of critical affine random walks, with a ping-pong argument played at a time-dependent place.