A fixed point theorem for the action of $SL_n$ over local fields on symmetric spaces of infinite dimension and finite rank

TL;DR

The paper establishes a fixed-point theorem for the action of $G=\mathrm{SL}_n(\mathbb{F})$ with $n\ge 3$ on infinite-dimensional symmetric spaces of finite rank over a non-archimedean local field. It develops a boundary-fixed-point strategy via harmonic, $\Gamma$-equivariant maps from the Bruhat–Tits building, supported by a spectral bound on link graphs and a convex energy framework. It then uses a Levi-type decomposition and an induction on rank, together with Property (FH) (equivalently Kazhdan’s Property (T)) and Property (F), to lift boundary fixed points to fixed points inside $X$, handling both finite- and infinite-dimensional target spaces. The results contribute to higher-rank rigidity phenomena for actions of $p$-adic groups on nonpositively curved, finite-rank spaces and provide a structural route combining harmonic maps, parabolic subgroups, and rank-based induction.

Abstract

Let F be a non-archimedean local field, and let $G = SL_n(F)$, $n \ge 3$. Let $X$ be an infinite-dimensional simply connected symmetric space of finite rank, with nonpositive curvature operator. We prove that every continuous action by isometries of $G$ on $X$ has a fixed point.

Theorems & Definitions (31)

• Definition 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 2.1
• Lemma 2.2
• proof