Piecewise automatic groups

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Entropy and isoperimetry for linear and non-linear group actions (2008, background): “(a) The existence of groups N Ë Z with arbitrarily fast growing Fı, where N is a locally finite group, is indicated in [9], and groups of intermediate growth with arbitrarily fast growing Fı are constructed in [10].”
Poisson-Furstenberg Boundaries, Large-scale Geometry and Growth of Groups (2011, methodology): “Moreover, one can use the construction from [35] to produce examples of groups H such that for any simple random walk onH it holds lim sup(logLH,μ(n))/n = 1, lim inf(logLH,μ(n))/n = 1/2.”
Milnor's Problem on the Growth of Groups and its Consequences (2011, methodology): “Oscillation techniques received further development in the paper of Erschler [76] (which was already mentioned at the end of section 12), where the notion of piecewise automatic group was introduced.”
Speed of random walks, isoperimetry and compression of finitely generated groups (2015, background): “itable assumptions, the metric in the diagonal product is essentially the same as in a direct product. The construction of diagonal products is reminiscent of the piecewise automata group of Erschler [Ers06] and the groups of Kassabov-Pak [KP13], which permit to obtain oscillating or "close to ”