Lifting subgroups of $\mathrm{PSL}_2$ to $\mathrm{SL}_2$ over local fields

TL;DR

The paper studies lifting discrete subgroups of $\mathrm{PSL}_2(K)$ to $\mathrm{SL}_2(K)$ over non-archimedean local fields. Using the Bruhat--Tits tree $T_K$ and Bass--Serre theory, it proves that, for $\mathrm{char}(K)\neq 2$, a discrete subgroup $G\le\mathrm{PSL}_2(K)$ lifts to $\mathrm{SL}_2(K)$ if and only if $G$ has no $2$-torsion, and it characterizes the structure of $G$ in characteristic $0$ as a free product of vertex stabilizers with a free group. The paper also constructs explicit non-lift examples when hypotheses are weakened and discusses lifting of representations through a cohomological lens, showing that vanishing of $H^2(\Gamma,\mathbb{Z}/2\mathbb{Z})$ guarantees liftability. Overall, the results extend Culler’s lifting phenomenon to non-archimedean settings and provide a non-archimedean analogue of Button’s obstructions, with implications for understanding subgroup dynamics and representations in $\mathrm{PSL}_2(K)$ and $\mathrm{SL}_2(K)$.

Abstract

Let $K$ be a non-archimedean local field. We show that discrete subgroups without 2-torsion in $\mathrm{PSL}_2(K)$ can always be lifted to $\mathrm{SL}_2(K)$, and provide examples (when $\mathrm{char}(K) \neq 2$) which cannot be lifted if either of these conditions is removed. We also briefly discuss lifting representations of groups into $\mathrm{PSL}_2(K)$ to $\mathrm{SL}_2(K)$.

Figures (1)

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Theorems & Definitions (33)

• Theorem A
• Theorem B
• Theorem C
• Remark 1.1
• Proposition D
• Proposition E
• Lemma 2.1
• Proposition 2.2
• Corollary 2.3
• Proposition 2.4