(3.6) As a first application, let us discuss the case of abelian varieties. Let $A$ be an abelian $K$-variety of dimension $n$, and denote by $\mathscr{A}$ its Néron model. Then Berkovich has constructed in [Be90, §6.5] a canonical skeleton $\Delta(A)$ in $A^{\mathrm{an}}$, together with a continuous retraction $\rho_{A}\colon A^{\mathrm{an}}\to\Delta(A)$, via the theory of non-archimedean uniformization. The dimension of $\Delta(A)$ is equal to the toric rank of $\mathscr{A}^{o}_{k}$ (the dimension of the maximal subtorus). Let us make this construction more precise in the maximally degenerate case. Assume that $A$ has purely toric reduction, that is, $\mathscr{A}^{o}_{k}$ is a torus. Let $e$ be the identity point on $A$. Then the universal pointed covering space of $(A,e)$ (with respect to the Berkovich topology) is isomorphic to the analytification of a split $n$-dimensional $K$-torus $T$. The kernel $L$ of the morphism $\pi\colon T^{an}\to A^{\mathrm{an}}$ is a lattice in $T(K)$ (called the period lattice), and the image $\rho_{T}(L)$ of $L$ in $N_{\mathbb{R}}$ is a lattice of rank $n$. By definition, the canonical skeleton $\Delta(A)$ is the image of $\Delta(T)$ under the map $\pi$. Moreover, we have a Cartesian diagram of topological spaces

such that $\rho_{A}$ sends $\Delta(A)$ homeomorphically onto $N_{\mathbb{R}}/\rho_{T}(L)$. In particular, $\Delta(A)$ is a real torus of dimension $n$, $\rho_{A}$ is an $n$-dimensional torus fibration, and the induced integral affine structure on $\Delta(A)$ coincides with the quotient structure on $N_{\mathbb{R}}/\rho_{T}(L)$.