Geodesic
Verbal subgroups of the finitary automorphism group of a $2$-adic tree
Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 3, pp. 875-888

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It is established that any verbal subgroup of the group of finitary automorphisms of a 2-adic rooted tree coincides with some term of its lower central series. The coincidences for derived subgroups, subgroups of $2^k$-th degrees and terms of the Engel series are indicated precisely. 
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     author = {N. V. Smetanyuk and V. I. Sushchanskii},
     title = {Verbal subgroups of the finitary automorphism group of a $2$-adic tree},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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N. V. Smetanyuk; V. I. Sushchanskii. Verbal subgroups of the finitary automorphism group of a $2$-adic tree. Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 3, pp. 875-888. http://geodesic.mathdoc.fr/item/FPM_2000_6_3_a17/