We prove that for a free product G with free factor system 𝒢, any automorphism ϕ preserving 𝒢, atoroidal (in a sense relative to 𝒢) and none of whose power send two different conjugates of subgroups in 𝒢 on conjugates of themselves by the same element, gives rise to a semidirect product G ⋊ϕℤ that is relatively hyperbolic with respect to suspensions of groups in 𝒢. We recover a theorem of Gautero–Lustig and Ghosh that, if G is a free group, ϕ an automorphism of G, and 𝒢 is its family of polynomially growing subgroups, then the semidirect product by ϕ is relatively hyperbolic with respect to the suspensions of these subgroups. We apply the first result to the conjugacy problem for certain automorphisms (atoroidal and toral) of free products of abelian groups.

中文翻译：

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自由积和自由群的自同构的相对双曲线

我们证明对于免费产品G自由因子系统𝒢, 任何自同构φ保存𝒢, atroidal (在某种意义上相对于𝒢) 并且没有一个权力将两个不同的子群共轭发送到𝒢由相同元素在它们自身的共轭上产生半直积G ⋊φℤ相对于组的悬浮而言是双曲线的𝒢. 我们恢复 Gautero-Lustig 和 Ghosh 的一个定理，如果G是一个自由组，φ的自同构G， 和𝒢是它的多项式增长子群族，然后是半直积φ相对于这些子群的悬浮而言是双曲线的。我们将第一个结果应用于阿贝尔群自由积的某些自同构（环形和环形）的共轭问题。