Subgaussian 1-cocycles on discrete groups

We prove the L_p Poincaré inequalities with constant Cp for 1-cocycles on countable discrete groups under Bakry-Émery's Γ₂-criterion. These inequalities determine an analog of subgaussian behavior for 1-cocycles. Our theorem improves some of our previous results in this direction, and in particular implies Efraim and Lust-Piquard's Poincaré-type inequalities for the Walsh system. The key new ingredient in our proof is a decoupling argument. As complementary results, we also show that the spectral gap inequality implies the L_p Poincaré inequalities with constant Cp under some conditions in the noncommutative setting. New examples which satisfy the Γ₂-criterion are provided as well.