Hierarchies for Relatively Hyperbolic Virtually Special Groups

TL;DR

The paper develops a relative-hyperbolic analogue of Wise's quasiconvex hierarchy framework for virtually compact special groups. By building CAT$(0)$ relative-hyperbolic pairs, proving a relative fellow traveling property, and constructing a Malnormal Quasiconvex Fully $ P$-elliptic hierarchy via the double dot and augmented cube complex techniques, it enables controlled Dehn fillings that preserve hyperbolicity and virtual specialness. The main contributions include a relative MSQT for arbitrary peripheral subgroups and a pathway to obtain malnormal, quasiconvex hierarchies terminating in peripheral groups, leading to new virtually special quotients in the relatively hyperbolic setting. This advances the understanding of relatively hyperbolic, virtually compact special groups and provides a robust toolkit for Dehn filling arguments in this context, with significant implications for constructing new virtually special groups and analyzing their peripheral structure.

Abstract

Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually compact special groups. We use this hierarchy to prove a generalization of Wise's Malnormal Special Quotient Theorem for relatively hyperbolic virtually compact special groups with arbitrary peripheral subgroups.

Figures (9)

• Figure 1: An example of a triangle which is $\delta$--thin relative to some $F$ with its comparison tripod. Points in the blue part of the tripod have preimages in the triangle which lie in the blue shaded region. All other points have preimages in the triangle with diameter $\delta$ like the point $p$ whose preimages $x,y$ have $d(x,y)<\delta$. The fat part (see Definition \ref{['Def: corner segments']}) of each side is the subsegment that intersects the blue shaded region.
• Figure 2: Applying the triangle inequality four times gives a bound on the difference between the length of $[p_{ab},p_{ba}]$ and the length of $[p_{bc},p_{cb}]$ in terms of $|[p_{ac},p_{ca}]|,\delta$.
• Figure 3: A graph of spaces realization of a genus 2 surface where $\Sigma_{1,1}$ is a punctured torus, together with the corresponding graph of groups obtained by applying the $\pi_1$ functor.
• Figure 4: A hierarchy for $\pi_1(\Sigma_2)$, the fundamental group of a genus $2$ surface $\Sigma_2$, where the iterated splitting of $\pi_1(\Sigma_2)$ cannot be realized by a graph of groups. The first splitting is over the infinite cyclic subgroup of $\pi_1(\Sigma_2)$ corresponding to one of the blue copies of $S^1$. The resulting vertex spaces are punctured tori whose fundamental groups are rank $2$ free groups. Cutting along the green arc in each punctured torus makes an annulus. Then the fundamental group of a punctured torus splits as an HNN extension of the fundamental group of an annulus ($\Z$) over the trivial group (corresponding to the green arcs in each annulus which are glued together to make a punctured torus).
• Figure 5: The quadrilateral constructed in the proof of Proposition \ref{['New rel hyp pair']}.

Theorems & Definitions (174)

• Definition 1.1: WiseManuscript
• Theorem 1.2: WiseManuscript, Wise's Quasiconvex Hierarchy Theorem
• Theorem 1.3: Wise's Malnormal Special Quotient Theorem WiseManuscript
• Theorem 0
• Theorem 0
• Definition 2.1: Hruska2010 Definition 3.6
• Proposition 2.2
• Corollary 2.3
• Definition 2.4
• Definition 2.5