临界指数小于1的可及CAT(-1)群
Accessible CAT$(-1)$ groups of critical exponent less than one
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中文总结 AI 辅助
该研究针对可及CAT(-1)群，证明临界指数小于1时群的结构性质，给出相关推论，覆盖更大类群并补充JSJ相关结论。
中文摘要 AI 辅助
设X是真CAT(-1)空间，Γ是Isom(X)中有限生成且离散的子群。核心结构定理表明：若Γ在有限子群上可及且δ_X(Γ)<1，则Γ是几何有限且几乎自由的，具有有限群图，其边群为有限群，无限顶点群为几乎循环群，且∂Γ→Λ_Γ恰好坍缩它们的共轭两点边界。该结论具有遗传性，指数低于1/2时，每个有限生成子群都是凸上有界的（见定理accessible-main）。因此，非几乎自由的可及群满足δ_X(Γ)≥1（见推论accessible-gap）。其推论涵盖有限表示群、有限子群阶一致有界的群、特征零线性群、Kleinian群，其中无限无抛物Kleinian群存在有限指数经典Schottky子群（见推论accessibility-extension、linear-groups、kleinian-classical）。由此覆盖的类远大于LiuWang2023、Hou2001的研究，另见注strictness-sharpness。最后，还给出其在有限JSJ代表元与分层上的推论（见定理JSJ、hierarchy，推论hierarchy-dimension）。
英文摘要
Let $X$ be proper CAT$(-1)$ and let $Γ\le\Isom(X)$ be finitely generated and discrete. The sharp structural theorem states that, if $Γ$ is accessible over finite subgroups and $δ_X(Γ)<1$, then $Γ$ is geometrically finite and virtually free, has a finite graph of groups with finite edge groups and virtually cyclic infinite vertex groups, and $\partialΓ\toΛ_Γ$ collapses exactly their conjugate two point boundaries. The result is hereditary, and below $1/2$ every finitely generated subgroup is convex-cobounded (\cref{thm:accessible-main}). Hence non-virtually-free accessible groups have $δ_X(Γ)\ge1$ (\cref{cor:accessible-gap}). Consequences cover finitely presented groups, groups with uniformly bounded finite-subgroup orders, characteristic-zero linear groups, and Kleinian groups, also infinite parabolic-free Kleinian groups have finite-index classical Schottky subgroups (\cref{cor:accessibility-extension,cor:linear-groups,cor:kleinian-classical}). Hence we cover substantial larger class than \cite{LiuWang2023},\cite{Hou2001}, also see \cref{rem:strictness-sharpness}. Finally, we also state consequences for finite JSJ representatives and hierarchies (\cref{thm:JSJ,thm:hierarchy,cor:hierarchy-dimension}).
