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

这是处理相对双曲群中的局部极限定理的两篇系列论文中的第一篇。在第一篇论文中，我们证明了格林函数的粗略估计。在此过程中，我们介绍了相对自动性的概念，这将在两篇论文中都很有用，并且我们证明了相对双曲群是相对自动的。我们还定义了相对双曲群上随机游走的谱正递归概念。然后我们使用我们对 Green 函数的估计来证明 $p_n\asymp R^{-n}n^{-3/2}$ 用于光谱正循环随机游走，其中 $p_n$ 是回到时间n的原点，其中R是随机游走的光谱半径的倒数。