Normal completions of toric varieties over rank one valuation rings and completions of $Γ$-admissible fans
We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $\Gamma$ is discrete or divisible then no base-change is needed. We give explicit examples which show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $\Gamma$-admissible fans admit $\Gamma$-admissible completions. In order to show this we prove a combinatorial analog of noetherian reduction which we believe will be of independent interest.