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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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