In this paper, we describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to the definition of the conjugacy length function. We show that for free solvable groups the conjugacy length function is at most cubic. For wreath products the behaviour depends on the conjugacy length function of the two groups involved, as well as subgroup distortion within the quotient group.
Funding source: EPSRC
The author would like to thank Cornelia Druţu for many valuable discussions on this paper. Alexander Olshanskii's comments on a draft copy were also very helpful, as were discussions with Romain Tessera. He would also like to thank Ralph Stöhr for providing a reference for Lemma 2.4 and an anonymous referee for helpful comments.
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