AI 中文总结
本研究针对无挠单端双曲群的自同构映射环面，通过典范JSJ树悬垂、Nielsen–Thurston约化细化及分块折叠得到余紧柱形分裂，将共轭问题归约为有理集判定，证明了这类循环扩张群的共轭问题可解。
AI 中文摘要
设$G$为无挠单端双曲群，$ϕ∈\operatorname{Aut}(G)$。我们证明映射环面（也称悬垂）$$M:=G_ϕ=G\rtimes_ϕ\mathbb{Z}$$具有可解的共轭问题。这一结果建立在Préaux的开创性工作基础上，后者解决了所有（可几何化的）三维流形的共轭问题。我们将$M$视为纤维化三维流形的推广。\n 我们的证明从$G$的典范JSJ树出发，其悬垂给出了$M$的群图分解。我们利用诱导单值化的Nielsen–Thurston约化系统细化每个悬垂QH顶点，同时考虑不可定向曲面与反向定向单值化。等价地，这是对每个为纤维化三维流形的顶点的进一步几何JSJ分解，但在分裂中既允许环面也允许克莱因瓶。\n 随后我们根据这些细化后的JSJ的片段是否共享中心元素，通过一系列折叠操作沿初等顶点粘合，将它们“分块”组合在一起。这种新的分裂是余紧且柱形的；特别地，某些局部“解集”被证明是几乎阿贝尔子群的有理子集——所涉及的几乎阿贝尔群是边群的直积。我们版本的Préaux群图论证随后将全局共轭问题归约为这些有理集的有效相交与非空性判定。
英文摘要
Let $G$ be a torsion-free one-ended hyperbolic group and let $ϕ\in\operatorname{Aut} (G)$. We prove that the mapping torus, or suspension, $$ M:=G_ϕ=G\rtimes_ϕ\mathbb{Z} $$ has solvable conjugacy problem. This builds on the pioneering work of Préaux who solved the conjugacy problem for all (geometrisable) three-manifolds. We view $M$ as a generalisation of a fibred three-manifold. Our proof begins with the canonical JSJ tree of $G$, whose suspension gives a graph-of-groups decomposition of $ M $. We refine each suspended QH vertex using a Nielsen--Thurston reduction system for its induced monodromy, taking account of non-orientable surfaces and orientation-reversing monodromy. Equivalently, this is a further geometric JSJ decomposition for each vertex which is a fibred three-manifold, but allowing Klein bottles as well as tori in the splitting. We then `block' pieces of this refined JSJ together according to whether they share a central element, glued along an elementary vertex, by a sequence of folding operations. This new splitting is cocompact and acylindrical; in particular certain local `solution sets' turn out to be rational subsets of virtually abelian subgroups - the virtually abelian groups in question are products of the edge groups. Our version of Préaux's graph-of-groups argument then reduces global conjugacy to effective intersection and non-emptiness for these rational sets.
CommentsUpdated to show that the block tree is automorphism invariant
