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

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 $\mathrm{char}(K)\ne 2$) 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 ${\mathrm{SL}}_2(K)$.

14 pages. v2: final version, to appear in Journal of Group Theory