  We prove structure theorems for o-minimal definable subsets $S\subset G$ of
definable groups containing large multiplicative structures, and show definable
groups do not have bounded torsion arbitrarily close to the identity. As an
application, for certain models of $n$-step random walks $X$ in $G$ we show
upper bounds $\mathbb{P}(X\in S)\le n^{-C}$ and a structure theorem for the
steps of $X$ when $\mathbb{P}(X\in S)\ge n^{-C'}$.
