Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:

• symmetric,
• adapted (in the sense that there is no proper subgroup $H$ such that $\mu(H)=1$),
• supported on a compact generating set of $G$,
• absolutely continuous w.r.t. the Haar measure on $G$.

Is there a "Poincaré inequality" for this settings?

If not, maybe for connected Lie group $G$?