Basmajian's identity over non-Archimedean local fields

Let $\mathrm{\Sigma }$ be a connected compact oriented surface with boundary and negative Euler characteristic. Let $k$ be a non-Archimedean local field. In this paper, we prove Basmajian’s identity for projective Anosov representations $\rho \text{\hspace{0.17em}⁣}:{\pi }_1\mathrm{\Sigma }\to \mathrm{P}\mathrm{S}\mathrm{L}(d,k),d\ge 2$. Our series identity exhibits a drastic difference from all the Basmajian-type identities over the Archimedean fields $\mathbb{R}$ and $\mathbb{C}$. In particular, the series is a signed finite sum. When $d=2$, we give a geometric proof of the identity using Berkovich hyperbolic geometry.