Conjugacy Length in Group Extensions

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behaviour of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products ${\mathbb{Z}}^d⋊{\mathbb{Z}}^k$ have at most exponential (if $k>1$) or linear (if $k=1$) conjugacy length functions.