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We prove that every linear-activity automaton group is amenable.  The proof is based on showing that a sufficiently symmetric random walk on aspecially constructed degree 1 automaton group -- the mother group -- hasasymptotic entropy 0.  Our result answers an open question by Nekrashevich in the Kourovka notebook,and gives a partial answer to a question of Sidki.

Amenability of linear-activity automaton groups