Chemical distance in graphs of polynomial growth

We prove an Antal-Pisztora type theorem for transitive graphs of polynomial growth. That is, we show that if $G$ is a transitive graph of polynomial growth and $p > p_c(G)$, then for any two sites $x, y$ of $G$ which are connected by a $p$-open path, the chemical distance from $x$ to $y$ is at most a constant times the original graph distance, except with probability exponentially small in the distance from $x$ to $y$. We also prove a similar theorem for general Cayley graphs of finitely presented groups, for $p$ sufficiently close to 1. Lastly, we show that all time constants for the chemical distance on the infinite supercritical cluster of a transitive graph of polynomial growth are Lipschitz continuous as a function of $p$ away from $p_c$.