Lifting subgroups of PSL₂ to SL₂ over local fields

Abstract Let 𝐾 be a non-archimedean local field. We show that discrete subgroups without 2-torsion in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PSL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{PSL}_{2}(K) can always be lifted to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}_{2}(K) , and provide examples (when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>char</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≠</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> \operatorname{char}(K)\neq 2 ) which cannot be lifted if either of these conditions is removed. We also briefly discuss lifting representations of groups into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PSL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{PSL}_{2}(K) to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}_{2}(K) .