AI 中文总结
该研究针对分层双曲群证明了拟平闭合与粗平环面定理，应用于考克斯特群的HHG结构判定及简化几乎可解子群的相关证明。
AI 中文摘要
我们证明了分层双曲群（HHG）的拟平闭合定理和粗平环面定理。即，给定一个HHG群G，我们证明G是双曲群当且仅当它不包含ℤ²子群；且若A≤G是几乎ℤⁿ群，则存在一个A-不变的n维一致质量拟平F，使得F中任意两点都可通过位于F内的一致质量分层路径连接。这一结论源于描述G中A的“粗极小集”的更详细定理，该定理有诸多应用，包括几乎阿贝尔子群的升链条件、最高阶阿贝尔子群的分层拟凸性，以及对阿贝尔子群的正规化子、中心化子和可公度子群的几何控制。我们利用这一点，根据某些考克斯特群的仿射子群排除其HHG结构，并给出HHG的几乎可解子群必为几乎阿贝尔群的新证明，该证明通过避开格罗莫夫的多项式增长定理简化了原证明。
英文摘要
We prove a quasiflat closing theorem and a coarse flat torus theorem for hierarchically hyperbolic groups (HHGs). Namely, given an HHG $G$, we prove that $G$ is hyperbolic if and only if it contains no $\mathbb Z^2$ subgroups and, if $A\leq G$ is virtually $\mathbb Z^n$, then there is an $A$--invariant $n$--dimensional uniform quality quasiflat $F$ such that any two points in $F$ are joined by a uniform-quality hierarchy path lying in $F$. The later is a consequence of a more detailed theorem describing a ``coarse minset'' for $A$ in $G$, which has various applications, including an ascending chain condition for virtually abelian subgroups, hierarchical quasiconvexity of highest abelian subgroups, and some geometric control over normalisers, centralisers, and commensurators of abelian subgroups. We use this to rule out HHG structures for certain Coxeter groups on the basis of their affine subgroups, and to give a new proof that virtually solvable subgroups of HHGs are virtually abelian, which simplifies the original proof by avoiding Gromov's polynomial growth theorem.
Comments65 pages, 1 figure
