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Subgroup distortion in wreath products of cyclic groups. (English) Zbl 1259.20049
Summary: We study the effects of subgroup distortion in the wreath products $$A\text{\,wr\,}\mathbb Z$$, where $$A$$ is finitely generated Abelian. We show that every finitely generated subgroup of $$A\text{\,wr\,}\mathbb Z$$ has distortion function equivalent to some polynomial. Moreover, for $$A$$ infinite, and for any polynomial $$l^k$$, there is a 2-generated subgroup of $$A\text{\,wr\,}\mathbb Z$$ having distortion function equivalent to the given polynomial. Also, a formula for the length of elements in arbitrary wreath product $$H\text{\,wr\,}G$$ easily shows that the group $$\mathbb Z_2\text{\,wr\,}\mathbb Z^2$$ has distorted subgroups, while the lamplighter group $$\mathbb Z_2\text{\,wr\,}\mathbb Z$$ has no distorted (finitely generated) subgroups. In the course of the proof, we introduce a notion of distortion for polynomials. We are able to compute the distortion of any polynomial in one variable over $$\mathbb Z$$, $$\mathbb R$$ or $$\mathbb C$$.

##### MSC:
 20F65 Geometric group theory 20E22 Extensions, wreath products, and other compositions of groups 20F05 Generators, relations, and presentations of groups 20E07 Subgroup theorems; subgroup growth 20F69 Asymptotic properties of groups
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