We prove the Lp 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 Lp Poincaré inequalities with constant Cp under some conditions in the noncommutative setting. New examples which satisfy the Γ2-criterion are provided as well.
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