paper of Ihara, why must eigenvalues of an element of a torsion-free cocompact lattice be in the ground field?

Question

In "Discrete Subgroups of PL(2,$k_{\mathcal{P}}$)" which appears on pp. 272-278 of "Algebraic Groups and Discontinuous Subgroups: Proceedings of Symposia in Pure Mathematics, Volume IX", Ihara makes the claim that if $\Gamma$ is a discrete torsion-free cocompact subgroup of PL(2,$K$) for a non-archimedean local field $K$, then the eigenvalues of any element of $\Gamma$ must be in $K$. I am not sure how to prove this, can anyone explain it to me?