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Let $G$ be a countable discrete group with an orthogonal representation$\alpha$ on a real Hilbert space $H$. We prove $L_p$ Poincar\'e inequalitiesfor the group measure space $L_\infty(\Omega_H,\gamma)\rtimes G$, where boththe group action and the Gaussian measure space $(\Omega_H, \gamma)$ areassociated with the representation $\alpha$. The idea of proof comes fromPisier's method on the boundedness of Riesz transform and Lust-Piquard's workon spin systems. Then we deduce a transportation type inequality from the $L_p$Poincar\'e inequalities in the general noncommutative setting. This inequalityis sharp up to a constant (in the Gaussian setting). Several applications aregiven, including Wiener/Rademacher chaos estimation and new examples ofRieffel's compact quantum metric spaces.

Poincar\'e type inequalities for group measure spaces and related transportation cost inequalities