Accessible CAT$(-1)$ groups of critical exponent less than one

Abstract

Let $X$ be proper CAT$(-1)$ and let $Γ\le\Isom(X)$ be finitely generated and discrete. The sharp structural theorem states that, if $\Gamma$ is accessible over finite subgroups and ${\delta }_X(\Gamma )<1$, then $\Gamma$ is geometrically finite and virtually free, has a finite graph of groups with finite edge groups and virtually cyclic infinite vertex groups, and $\partial \Gamma \to {\Lambda }_{\Gamma }$ 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 ${\delta }_X(\Gamma )\ge 1$ (\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}).