Abstract: In this talk we consider the centred and the uncentred Hardy--Littlewood maximal operators -- denoted $\cM$ and $\cN$, respectively -- on certain metric measure spaces with exponential volume growth and ``locally bounded geometry'' such as the hyperbolic upper half-plane and homogeneous trees. A classical result of Hardy and Littlewood states that on the Euclidean space both $\cM$ and $\cN$ are bounded on $L^p$ for all $p>1$ and that…