Abstract
The Lamplighter group L 2 can be realized in different ways. We construct several groups whose elements are very different, yet which can be considered the same group, L 2, because they are isomorphic and they can all be presented in the same way. We give a description of L 2 as a dynamical system, as a group using an infinite direct sum in its definition and as a self-similar group generated by a 2-state automaton, as shown by R. Grigorchuk and A. Żuk.
It may well be that this man is absurd. But he is not so absurd as the king, the conceited man, the businessman, and the tippler. For at least his work has some meaning. When he lights his street lamp, it is as if he brought one more star to life, or one flower. When he puts out his lamp, he sends the flower, or the star, to sleep. That is a beautiful occupation. And since it is beautiful, it is truly useful.
Antoine de Saint-Exupéry
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Bonanome, M.C., Dean, M.H., Dean, J.P. (2018). The Lamplighter Group L2. In: A Sampling of Remarkable Groups. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01978-5_4
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DOI: https://doi.org/10.1007/978-3-030-01978-5_4
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