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The width of a power of a free nilpotent group of nilpotency class $$2$$. (English. Russian original) Zbl 0956.20028
Sib. Math. J. 41, No. 1, 173-179 (2000); translation from Sib. Mat. Zh. 41, No. 1, 206-213 (2000).
Let $$G$$ be a group. Let $$wG$$ be the verbal subgroup of $$G$$ defined by a word $$w$$. Recall that the width $$\text{wid}(g,w)$$ of $$g\in wG$$ with respect to $$w$$ is the least number $$l$$ such that $$g$$ is equal to a product of $$l$$ values of $$w^{\pm 1}$$ in $$G$$. For a subset $$M$$ of $$wG$$, define $$\text{wid}(g,M)$$ by $$\max_{g\in M}\text{wid}(g,w)$$.
The author proves that the width of a free nilpotent finitely generated group of class 2 with respect to $$x^k$$ is finite and does not depend on $$k$$. More precisely, the following theorem is proven: Let $$N_n$$ be a free nilpotent group of class 2 and let $$k\geq 1$$ be a positive integer. Then (1) $$\text{wid}(x^{2k},N_n^{2k})=2[n/2]+1$$, where $$n\geq 2$$; (2) $$\text{wid}(x^{2k+1},N_n^{2k+1})=1$$.

##### MSC:
 20F18 Nilpotent groups 20F05 Generators, relations, and presentations of groups 20F12 Commutator calculus 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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