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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 \(\mathbb C\).

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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