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 "Poincare inequality" for this settings? 

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