There are several examples where the mixing time of a Markov chain can be reduced substantially, often to about its square root, by "lifting", i.e., by splitting each state into several states.In several examples of random walks on groups, the lifted chain not only mixes better, but is easier to analyze.We characterize the best mixing time achievable through lifting in terms of multicommodity flows. We show that the reduction to square root is best possible. If the lifted chain is time-reversible, then the gain is smaller, at most a factor of log(l/na), where 110 is the smallest stationary probability of any state. We give an example showing that a gain of a factor of log(l/~o)/log log(l/rro) is possible.

A sequence in the additive group Z n of integers modulo n is called n-zero-free if it does not contain subsequences with length n and sum zero. The article characterizes the n-zero-free sequences in Z n of length greater than 3n/2−1. The structure of these sequences is completely determined, which generalizes a number of previously known facts. The characterization cannot be extended in the same form to shorter sequence lengths. Consequences of the main result are best possible lower bounds for the maximum multiplicity of a term in an n-zero-free sequence of any given length greater than 3n/2−1 in Z n , and also for the combined multiplicity of the two most repeated terms. Yet another application is finding the values in a certain range of a function related to the classic theorem of Erdős, Ginzburg and Ziv.

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