Effective bounds for hierarchies and the Malnormal Special Quotient Theorem for relatively hyperbolic virtually special groups

Einstein (Algebraic & Geometric Topology 25 (2025) 4437–4497; arXiv:1903.12284) proved that a relatively hyperbolic group pair $(G,\mathcal{P})$ with $G$ virtually compact special has a finite-index subgroup $G_0$ carrying a quasiconvex, malnormal, fully elliptic hierarchy (Theorem 1), and deduced a relatively hyperbolic Malnormal Special Quotient Theorem (Theorem 2): there exist finite-index subgroups $\dot{P}_i\triangleleft P_i$ such that every peripherally finite Dehn filling with kernels $N_i\triangleleft\dot{P}_i$ yields a hyperbolic, virtually special quotient. Both existence proofs are non-effective. The open problem is their effective, quantitative version: given an effective description of the pair, is there a provably terminating algorithm that outputs $G_0$ and the subgroups $\dot{P}_i$, and is there a computable explicit bound on the indices $[G:G_0]$ and $[P_i:\dot{P}_i]$ — or does no such uniform algorithm or computable bound exist?

Background

A finitely generated group is Gromov-hyperbolic if geodesic triangles in its Cayley graph are uniformly δ-thin; intuitively, the group looks negatively curved at large scale. A group pair $(G,\mathcal{P})$ consists of a group $G$ with a finite collection $\mathcal{P}=\{P_1,\dots,P_m\}$ of finitely generated subgroups, called peripheral subgroups; the pair is relatively hyperbolic if $G$ is hyperbolic 'outside' its peripheral subgroups, the algebraic model being a finite-volume hyperbolic manifold with cusps, whose fundamental group is hyperbolic relative to the cusp subgroups (each isomorphic to $\mathbb{Z}^2$ in dimension 3). A group-theoretic Dehn filling of $(G,\mathcal{P})$ is a quotient $G(N_1,\dots,N_m)=G/\langle\!\langle\bigcup_i N_i\rangle\!\rangle$ obtained by killing subgroups $N_i\le P_i$ (the filling kernels); the filling is peripherally finite if each $N_i$ has finite index in $P_i$, and a property holds for all sufficiently long fillings if there is a finite set $B\subseteq G\setminus\{1\}$ such that every filling with $N_i\cap B=\varnothing$ for all $i$ has the property. Osin proved that all sufficiently long fillings of a relatively hyperbolic pair are relatively hyperbolic (with controlled kernel); Groves–Manning gave an independent geometric approach via the cusped space. A subgroup $H\le G$ is malnormal if $H\cap H^g=\{1\}$ for $g\notin H$, and quasiconvex (in the hyperbolic case) if its orbits stay uniformly close to geodesics. A CAT(0) cube complex is a simply connected complex of unit cubes glued along faces so that the link condition forces nonpositive curvature; it comes with canonical codimension-one subspaces called hyperplanes. Haglund–Wise defined a compact cube complex to be special if it admits a locally isometric immersion into the Salvetti complex (its standard classifying cube complex) of a right-angled Artin group — the group presented by a graph whose vertices are generators and whose edges are commuting relations — so that in particular its fundamental group embeds in that right-angled Artin group; a group is virtually compact special if it has a finite-index subgroup that is the fundamental group of a compact special cube complex. Wise's Quasiconvex Hierarchy Theorem characterizes hyperbolic virtually compact special groups as those splitting iteratively as graphs of groups over quasiconvex subgroups (a quasiconvex hierarchy) with finite terminal groups; this was the engine of Agol's proof of the Virtual Haken Conjecture. A key companion is Wise's Malnormal Special Quotient Theorem (MSQT): filling a hyperbolic virtually special group along an almost malnormal collection of quasiconvex subgroups preserves hyperbolicity and virtual specialness for a 'positive fraction' of all fillings, meaning there exist finite-index subgroups $\dot{P}_i\le P_i$ such that every filling with kernels contained in (respectively normal in) the $\dot{P}_i$ works; agol–Groves–Manning gave an independent proof. Einstein extended both results to the relatively hyperbolic setting with arbitrary peripheral subgroups: Theorem 1 provides, after passage to a finite-index subgroup $G_0\le G$, a quasiconvex, malnormal hierarchy that is fully elliptic (each conjugate of a peripheral subgroup is conjugate into a vertex group and, for every edge group $E$, either meets $E$ finitely or is contained in $E$, so peripheral subgroups are never cut up by the splittings; see Einstein, Definition 3.15), and Theorem 2 provides subgroups $\dot{P}_i\triangleleft P_i$ of finite index such that every peripherally finite filling with kernels $N_i\triangleleft\dot{P}_i$ is hyperbolic and virtually special. Both proofs are existence proofs and are non-effective in two distinct places. First, several finite covers and finite-index subgroups are obtained from residual finiteness (separability) arguments, which guarantee existence while giving no explicit bound on the index; one step, the 'double dot cover' (the cover of a cube complex corresponding to the kernel of the mod-2 hyperplane-intersection map into $(\mathbb{Z}/2)^{|\mathcal{W}|}$ over the non-separating hyperplanes $\mathcal{W}$), is elementary and has degree at most $2^{|\mathcal{W}|}$, but the separability steps have no known quantitative bound. Second, the Dehn-filling steps invoke Osin's theorem as a black box, supplying no computable finite exclusion set $B$ — something needed even to choose the filling kernels in the proof of Theorem 2. Consequently, no algorithm is known that produces $G_0$ or the $\dot{P}_i$ from an effective description of the pair, and no explicit bound on their indices is known. Conversely, one cannot hope to weaken the normality hypothesis itself: Lubotzky–Manning–Wilton refuted the stronger statement in which all sufficiently long peripherally finite fillings are required to be virtually special, already for a free base group. So the genuinely open question about Einstein's theorems is their effectivity, and that is the problem recorded here.

Problem Statement

Let $(G,\mathcal{P})$ with $\mathcal{P}=\{P_1,\dots,P_m\}$ be a relatively hyperbolic group pair in which $G$ is virtually compact special, given by effective input data: a finite presentation of $G$, a finite relative generating presentation with finite generating sets for each $P_i$, and a compact special cube complex $X$, supplied with its specialness data (the locally isometric immersion into the Salvetti complex of a right-angled Artin group) and with the homomorphism $\pi_1(X)\to G$, given by words in the generators of $G$, whose image has finite index in $G$ — so that the input itself witnesses the hypothesis that $G$ is virtually compact special. Einstein's Theorem 1 guarantees a finite-index subgroup $G_0\le G$ carrying a quasiconvex, malnormal, fully elliptic hierarchy, and Theorem 2 guarantees finite-index subgroups $\dot{P}_i\triangleleft P_i$ such that every peripherally finite Dehn filling with kernels $N_i\triangleleft\dot{P}_i$ is hyperbolic and virtually special. The problem, in two tiers: (1) Effectivity. Is there an algorithm which, on every input as above, provably terminates and outputs such a subgroup $G_0$ together with generating sets, and subgroups $\dot{P}_i$ satisfying the conclusion of Theorem 2? (2) Quantitative bounds. Is there a computable function of the input description bounding the indices $[G:G_0]$ and $[P_i:\dot{P}_i]$? The two tiers are separate questions, each either positively resolvable (for tier (1), a provably terminating algorithm with a proof of termination; for tier (2), an explicit computable bound with proof) or negatively resolvable (a proof that no such uniform algorithm, respectively no such computable bound, exists); a mixed outcome — for instance, an explicit bound together with a proof that no uniform terminating algorithm exists — is possible.

References

1. Eduard Einstein, Hierarchies for relatively hyperbolic virtually special groups, Algebraic & Geometric Topology 25 (2025) 4437–4497. DOI: 10.2140/agt.2025.25.4437, arXiv:1903.12284. https://doi.org/10.2140/agt.2025.25.4437
2. Alexander Lubotzky, Jason Fox Manning, Henry Wilton, Generalized triangle groups, expanders, and a problem of Agol and Wise, Commentarii Mathematici Helvetici 94 (2019), no. 1, 53–66. DOI: 10.4171/cmh/454, arXiv:1702.08200. https://doi.org/10.4171/cmh/454
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