Basmajian's identity over non-Archimedean local fields

TL;DR

This paper extends Basmajian's identity to non-Archimedean local fields for projective Anosov representations, revealing a finite signed sum series and providing a geometric proof in the case of $d=2$ using Berkovich hyperbolic geometry.

Contribution

It proves Basmajian's identity over non-Archimedean fields for projective Anosov representations, introducing a finite signed sum series and a geometric proof for $d=2$.

Findings

Series is a signed finite sum, unlike Archimedean cases.

Provides geometric proof for $d=2$ using Berkovich hyperbolic geometry.

Reveals a drastic difference from classical identities over $ eal$ and $bc$.

Abstract

Let $\Sigma$ 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 $\rho \colon \pi_1\Sigma \to {\rm PSL}(d,k), d\ge 2$. 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.