Accessible CAT(−1) groups of critical exponent less than one
Abstract
Let X be proper CAT(−1) and let Γ≤Isom(X) be finitely generated and discrete. The sharp structural theorem states that, if Γ is accessible over finite subgroups and δX(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 ∂Γ→ΛΓ 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 δX(Γ)≥1 (\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}).
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BibTeX
@misc{hou2026accessible,
title = {{Accessible CAT$(-1)$ groups of critical exponent less than one}},
author = {Yong Hou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00557},
eprint = {2609.00557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00557},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Accessible CAT(−1) groups of critical exponent less than one
AU - Yong Hou
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00557
UR - https://arxiv.org/abs/2609.00557
AB - Let X be proper CAT(−1) and let Γ≤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 ∂Γ→Λ_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)≥1 (\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}).
ER -