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This article examines the rate of escape for a random walk on G Z and proves laws of the iterated logarithm for both the inner and outer radius of escape. The class of G for which these results hold includes finite, G as well as groups of the form H Z, so the construction can be iterated. Laws of the iterated logarithm are also found for random walk on Baumslag-Solitar groups and a discrete version of the Sol geometry.doi:10.1214/aop/1068646371 fatcat:qua7q5g24zahnd6iimoxvcrrui