Projection and fibering in groups of bounded exponent

We develop a projection and fibering method for sets of small combinatorial doubling in (not necessarily abelian) discrete groups. As an application, in the abelian case we prove that, if $A$ is finite, the ambient group has exponent $r$, and $\mathrm{\mid }A+A\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }$, then $$\mathrm{\mid }⟨A⟩\mathrm{\mid }\le r^{(2+o(1))K}\mathrm{\mid }A\mathrm{\mid }\mathrm{.}$$ This answers a question of Ruzsa with an optimal leading coefficient, independent of Fox–Pham. The main ingredient is a discrete version of a fiber spillover argument. For sets in $2$-step nilpotent groups of exponent $r$, we also prove that $\mathrm{\mid }{A}^3\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }$ implies $\mathrm{\mid }⟨A⟩\mathrm{\mid }\le r^{(2+o_K(1))K}\mathrm{\mid }A\mathrm{\mid }$. The proof combines the abelian theorem with a weighted averaging of central fibers and commutators.