  We prove the $L_p$ Poincar\'e inequalities with constant $C\sqrt{p}$ for
$1$-cocycles on countable discrete groups under Bakry--Emery's
$\Gamma_2$-criterion. These inequalities determine an analogue 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 Poincar\'e
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 $L_p$ Poincar\'e inequalities with constant $C{p}$
under some conditions in the noncommutative setting. New examples which satisfy
the $\Gamma_2$-criterion are provided as well.
