Local limit theorems in relatively hyperbolic groups I: rough estimates

Abstract This is the first of a series two papers dealing with local limit theorems in relatively hyperbolic groups. In this paper, we prove rough estimates for Green function. Along way, introduce notion relative automaticity which will be useful both and show that groups are automatic. We also define spectral positive recurrence random walks on then use our function to $p_n\asymp R^{-n}n^{-3/2}$ spectrally positive-recurrent walks, where $p_n$ probability going back origin at time n R inverse radius walk.