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QUANTIFYING SEPARABILITY IN VIRTUALLY SPECIAL GROUPS MARKF.HAGENANDPRIYAMPATEL Abstract. Wegiveanew, effectiveproofoftheseparabilityofcubicallyconvex-cocompactsubgroupsof specialgroups. Asaconsequence, weshowthatifGisavirtuallycompactspecialhyperbolicgroup, and Q ≤ GisaK-quasiconvexsubgroup,thenanyg ∈ G−Qofword-lengthatmostnisseparatedfromQ byasubgroupwhoseindexispolynomialinnandexponentialinK.ThisgeneralizesaresultofBou-Rabee 6 andtheauthorsonresidualfinitenessgrowth[9]andaresultofthesecondauthoronsurfacegroups[34]. 1 0 2 b e Introduction F Early motivation for studying residual finiteness and subgroup separability was a result of the rel- 5 evance of these properties to decision problems in group theory. An observation of Dyson [18] and ] Mostowski [31], related to earlier ideas of McKinsey [30], states that finitely presented residually finite R groupshavesolvablewordproblem. Thewordproblemisaspecialcaseofthemembershipproblem, i.e. G theproblemofdeterminingwhetheragiveng ∈ GbelongstoaparticularsubgroupH ofG. Separability h. canproduceasolutiontothemembershipprobleminessentiallythesamewaythatasolutiontotheword t problemisprovidedbyresidualfiniteness(see,e.g.,thediscussionin[2]). AsubgroupH ≤ Gisseparable a m inGif,forallg ∈ G−H,thereexistsG(cid:48) ≤ GwithH ≤ G(cid:48) andg (cid:54)∈ G(cid:48). Producinganupperbound,in f.i. [ termsofgandH,ontheminimalindexofsuchasubgroupG(cid:48)iswhatwemeanbyquantifyingseparability ofH inG. Quantifyingseparabilityisrelatedtothemembershipproblem;seeRemarkDbelow. 2 Recently,separabilityhasplayedacrucialroleinlow-dimensionaltopology,namelyintheresolutions v 1 oftheVirtuallyHakenandVirtuallyFiberedconjectures[1,40]. Itsinfluenceintopologyisaconsequence 0 oftheseminalpaperofScott[37],whichestablishesatopologicalreformulationofsubgroupseparability. 0 Roughly, Scott’s criterion allows one to use separability to promote (appropriately construed) immer- 7 0 sions to embeddings in finite covers. In [1], Agol proved the Virtually Special Conjecture of Wise, an . outstanding component of the proofs of the above conjectures. Agol’s theorem shows that every word 1 0 hyperbolic cubical group virtually embeds in a right-angled Artin group (hereafter, RAAG). Cubically 5 convex-cocompact subgroups of RAAGs are separable [25, 22] and Agol’s theorem demonstrates that 1 word hyperbolic cubical groups inherit this property via the virtual embeddings (separability properties : v are preserved under passing to subgroups and finite index supergroups). In fact, since quasiconvex sub- i X groups of hyperbolic cubical groups are cubically convex-cocompact [22, 36], all quasiconvex subgroups r ofsuchgroupsareseparable. Inthispaper, wegiveanew, effectiveproofoftheseparabilityofcubically a convex-cocompactsubgroupsofspecialgroups. Ourmaintechnicalresultis: TheoremA. LetΓbeasimplicialgraphandletZ beacompactconnectedcubecomplex,basedata0–cube x,withabasedlocalisometryZ → S . Forallg ∈ A −π Z,thereisacubecomplex(Y,x)suchthat: Γ Γ 1 (1) Z ⊂ Y; (2) thereisabasedlocalisometryY → S suchthatZ → S factorsasZ (cid:44)→ Y → S ; Γ Γ Γ (3) anyclosedbasedpathrepresentingg liftstoanon-closedpathatxinY; (4) |Y(0)| ≤ |Z(0)|(|g|+1), where|g|isthewordlengthofg withrespecttothestandardgenerators. Date:February8,2016. 2010MathematicsSubjectClassification. 20E26,20F36. 1 Via Haglund-Wise’s canonical completion [23], Theorem A provides the following bounds on the sepa- rabilitygrowthfunction(definedinSection1)oftheclassofcubicallyconvex-cocompactsubgroupsofa (virtually) special group. Roughly, separability growth quantifies separability of all subgroups in a given class. CorollaryB. LetG ∼= π X,withX acompactspecialcubecomplex,andletQ betheclassofsubgroups 1 R representedbycompactlocalisometriestoX whosedomainshaveatmostRvertices. Then SepQR(Q,n) ≤ PRn G,S forallQ ∈ Q andn ∈ N,wheretheconstantP dependsonlyonthegeneratingsetS. Hence,lettingQ(cid:48) be R K theclassofsubgroupsQ ≤ GsuchthattheconvexhullofQx˜liesinNK(Qx˜)andx˜ ∈ X(cid:101)(0), SepQ(cid:48)K(Q,n) ≤ P(cid:48)gr (K)n, G,S X(cid:101) whereP(cid:48) dependsonlyonG,X(cid:101),S,andgr isthegrowthfunctionofX(cid:101)(0). X(cid:101) Inthehyperboliccase,wherecubicallyconvex-cocompactnessisequivalenttoquasiconvexity,weobtain aboundthatispolynomialinthelengthofthewordandexponentialinthequasiconvexityconstant: Corollary C. Let G be a group with an index-J special subgroup. Fixing a word-length (cid:107) − (cid:107) on G, S suppose that (G,(cid:107) − (cid:107) ) is δ-hyperbolic. For each K ≥ 1, let Q be the set of subgroups Q ≤ G such S K thatQisK-quasiconvexwithrespectto(cid:107)−(cid:107) . ThenthereexistsaconstantP = P(G,S)suchthatforall S K ≥ 0,Q ∈ Q ,andn ≥ 0, K SepQK(Q,n) ≤ Pgr (PK)J!nJ!, G,S G wheregr isthegrowthfunctionofG. G CorollaryCsaysthatifGisahyperboliccubicalgroup,Q ≤ GisK-quasiconvex,andg ∈ G−Q,then g isseparatedfromQbyasubgroupofindexboundedbyafunctionpolynomialin(cid:107)g(cid:107) andexponential S inK. The above results fit into a larger body of work dedicated to quantifying residual finiteness and sub- groupseparabilityofvariousclassesofgroups(see,e.g.,[10,11,27,14,34,33,35,8,12,29]). WhenGisthe fundamentalgroupofahyperbolicsurface,compareCorollaryCto[34,Theorem7.1]. Combiningvarious cubulationresultswith[1],thegroupscoveredbyCorollaryCincludefundamentalgroupsofhyperbolic 3–manifolds [6, 26], hyperbolic Coxeter groups [24], simple-type arithmetic hyperbolic lattices [5], hy- perbolicfree-by-cyclicgroups[21],hyperbolicascendingHNNextensionsoffreegroupswithirreducible monodromy [20], hyperbolic groups with a quasiconvex hierarchy [40], C(cid:48)(1) small cancellation groups 6 [38],andhencerandomgroupsatlowenoughdensity[32],amongmanyothers. In[9]Bou-Rabeeandtheauthorsquantifiedresidualfinitenessforvirtuallyspecialgroups,byworking in RAAGs and appealing to the fact that upper bounds on residual finiteness growth are inherited by finitely-generated subgroups and finite-index supergroups. Theorem A generalizes a main theorem of [9], andaccordinglytheproofisreminiscentoftheonein[9]. However, residualfinitenessisequivalent to separability of the trivial subgroup, and thus it is not surprising that quantifying separability for an arbitrary convex-cocompact subgroup of a RAAG entails engagement with a more complex geometric situation. Ourtechniquesthussignificantlygeneralizethoseof[9]. RemarkD(Membershipproblem). IfHisafinitely-generatedseparablesubgroupofthefinitely-presented group G, and one has an upper bound on Sep{H}(|g|), for some finite generating set S of G, then the G,S following procedure decides if g ∈ H: first, enumerate all subgroups of G of index at most Sep{H}(|g|) G,S usingafinitepresentationofG. Second,foreachsuchsubgroup,testwhetheritcontainsg;ifso,ignoreit, andifnot,proceedtothethirdstep. Third,foreachfinite-indexsubgroupnotcontainingg,testwhether 2 itcontainseachofthefinitelymanygeneratorsofH;ifso,wehaveproducedafinite-indexsubgroupcon- taining H but not g, whence g (cid:54)∈ H. If we exhaust the subgroups of index at most Sep{H}(|g|) without G,S finding such a subgroup, then g ∈ H. In particular, Corollary C gives an effective solution to the mem- bershipproblemforquasiconvexsubgroupsofhyperboliccubicalgroups,thoughitdoesnotappeartobe anymoreefficientthanthemoregeneralsolutiontothemembershipproblemforquasiconvexsubgroups of(arbitrary)hyperbolicgroupsrecentlygivenbyKharlampovich–Miasnikov–Weilin[28]. The paper is organized as follows: In Section 1, we define the separability growth of a group with respecttoaclassQofsubgroups,whichgeneralizestheresidualfinitenessgrowthintroducedin[7]. We alsoprovidesomenecessarybackgroundonRAAGsandcubicalgeometry. InSection2,wediscusscorol- lariestothemaintechnicalresult,includingCorollaryC,beforeconcludingwithaproofofTheoremAin Section3. Acknowledgments. We thank K. Bou-Rabee, S. Dowdall, F. Haglund, D.B. McReynolds, N. Miller, H. Wilton,andD.T.Wiseforhelpfuldiscussionsaboutissuesrelatedtothispaper. Wealsothankananony- mous referee for astute corrections. The authors acknowledge travel support from U.S. National Science FoundationgrantsDMS1107452,1107263,1107367"RNMS:GeometricStructuresandRepresentationVa- rieties"(theGEARNetwork)andfromgrantNSF1045119. M.F.H.wassupportedbytheNationalScience FoundationunderGrantNumberNSF1045119. 1. Background 1.1. Separability growth. Let G be a group generated by a finite set S and let H ≤ G be a subgroup. LetΩ = {∆ ≤ G : H ≤ ∆},anddefineamapDΩH : G−H → N∪{∞}by H G DΩH(g) = min{[G : ∆] : ∆ ∈ Ω ,g (cid:54)∈ ∆}. G H Thisisaspecialcaseofthenotionofadivisibilityfunctiondefinedin[7]anddiscussedin[11]. Notethat H isaseparablesubgroupofGifandonlyifDΩH takesonlyfinitevalues. G The separability growth of G with respect to a class Q of subgroups is a function SepQ : Q×N → G,S N∪{∞}givenby (cid:110) (cid:111) SepQ (Q,n) = max DΩQ(g) : g ∈ G−Q,(cid:107)g(cid:107) ≤ n . G,S G S If Q is a class of separable subgroups of G, then the separability growth measures the index of the subgroup to which one must pass in order to separate Q from an element of G − Q of length at most n. For example, when G is residually finite and Q = {{1}}, then SepQ is the residual finiteness growth G,S function. The following fact is explained in greater generality in [9, Section 2]. (In the notation of [9], SepQ (Q,n) = RFΩQ (n)forallQ ∈ Qandn ∈ N.) G,S G,S Proposition 1.1. Let G be a finitely generated group and let Q be a class of subgroups of G. If S,S(cid:48) are finitegeneratingsetsofG,thenthereexistsaconstantC > 0with SepQ (Q,n) ≤ C ·SepQ (Q,Cn) G,S(cid:48) G,S forQ ∈ Q,n ∈ N. Hencetheasymptoticgrowthrateof SepQ isindependentofS. G,S (Similar statements assert that upper bounds on separability growth are inherited by finite-index super- groupsandarbitraryfinitely-generatedsubgroupsbutwedonotuse,andthusomit,these.) 3 1.2. Nonpositively-curved cube complexes. We assume familiarity with nonpositively-curved and CAT(0) cube complexes and refer the reader to e.g. [19, 22, 39, 40] for background. We now make explicit some additional notions and terminology, related to convex subcomplexes, which are discussed in greater depth in [4]. We also discuss some basic facts about RAAGs and Salvetti complexes. Finally, we willusethemethodofcanonicalcompletion,introducedin[23],andreferthereaderto[9,Lemma2.8]for theexactstatementneededhere. 1.2.1. Local isometries, convexity, and gates. A local isometry φ : Y → X of cube complexes is a locally injectivecombinatorialmapwiththepropertythat,ife ,...,e are1–cubesofY allincidenttoa0–cube 1 n y,andthe(necessarilydistinct)1–cubesφ(e ),...,φ(e )alllieinacommonn–cubec(containingφ(y)), 1 n then e ,...,e span an n–cube c(cid:48) in Y with φ(c(cid:48)) = c. If φ : Y → X is a local isometry and X is 1 n nonpositively-curved, then Y is as well. Moreover, φ lifts to an embedding φ˜ : Y(cid:101) → X(cid:101) of universal covers,andφ˜(Y(cid:101))isconvexinX(cid:101) inthefollowingsense. LetX(cid:101) beaCAT(0)cubecomplex. ThesubcomplexK ⊆ X(cid:101) isfullifKcontainseachn–cubeofX(cid:101) whose 1-skeletonappearsinK. IfK isfull,thenK isisometricallyembeddedifK∩(cid:84) H isconnectedwhenever i i {Hi} is a set of pairwise-intersecting hyperplanes of X(cid:101). Equivalently, the inclusion K(1) (cid:44)→ X(cid:101)(1) is an isometric embedding with respect to the graph-metric. If the inclusion K (cid:44)→ X(cid:101) of the full subcomplex K is a local isometry, then K is convex. Note that a convex subcomplex is necessarily isometrically embedded,andinfactK isconvexifandonlyifK(1) ismetricallyconvexinX(cid:101)(1). Aconvexsubcomplex K is a CAT(0) cube complex in its own right, and its hyperplanes have the form H ∩K, where K is a hyperplaneofX(cid:101). Moreover,ifK isconvex,thenhyperplanesH1∩K,H2∩K ofK intersectifandonly if H ∩ H (cid:54)= ∅. We often say that the hyperplane H crosses the convex subcomplex K to mean that 1 2 H ∩K (cid:54)= ∅andwesaythehyperplanesH,H(cid:48) crossiftheyintersect. Hyperplanesareanimportantsourceofconvexsubcomplexes,intworelatedways. First,recallthatfor all hyperplanes H of X(cid:101), the carrier N(H) is a convex subcomplex. Second, N(H) ∼= H ×[−1, 1], and 2 2 the subcomplexes H ×{±1} of X(cid:101) “bounding” N(H) are convex subcomplexes isomorphic to H (when 2 H isgiventhecubicalstructureinwhichitsn–cubesaremidcubesof(n+1)–cubesofX(cid:101)). Asubcomplex of the form H × {±1} is a combinatorial hyperplane. The convex hull of a subcomplex S ⊂ X(cid:101) is the 2 intersectionofallconvexsubcomplexesthatcontainS;see[22]. LetK ⊆ X(cid:101) beaconvexsubcomplex. ThenthereisamapgK : X(cid:101)(0) → K suchthatforallx ∈ X(cid:101)(0), thepointg (x)istheuniqueclosestpointofK tox. (ThispointisoftencalledthegateofxinK;gates K arediscussedfurtherin[17]and[3].) ThismapextendstoacubicalmapgK : X(cid:101) → K,thegatemap. See e.g.[4]foradetaileddiscussionofthegatemapinthelanguageusedhere;weuseonlythatitextendsthe map on0–cubes andhas the property thatfor allx,y, ifg (x),g (y)are separatedby a hyperplaneH, K K thenthesameH separatesxfromy. Finally, thehyperplaneH separatesxfromg (x)ifandonlyifH K separates x from K. The gate map allows us to define the projection of the convex subcomplex K(cid:48) of X(cid:101) ontoK tobeg (K),whichistheconvexhulloftheset{g (x) ∈ K : x ∈ K(cid:48)(0)}. Convexsubcomplexes K(cid:48) K K,K(cid:48) are parallel if g (K) = K(cid:48) and g (K(cid:48)) = K. Equivalently, K,K(cid:48) are parallel if and only if, for K(cid:48) K eachhyperplaneH,wehaveH ∩K (cid:54)= ∅ifandonlyifH ∩K(cid:48) (cid:54)= ∅. Notethatparallelsubcomplexesare isomorphic. Remark1.2. Weoftenusethefollowingfacts. LetK,K(cid:48)beconvexsubcomplexesofX(cid:101). Thentheconvex hullC ofK ∪K(cid:48) containstheunionofK,K(cid:48) andaconvexsubcomplexoftheformG (K(cid:48))×γˆ,where K G (K(cid:48)) is the image of the gate map discussed above and γˆ is the convex hull of a geodesic segment γ K joining a closest pair of 0–cubes in K,K(cid:48), by [4, Lemma 2.4]. A hyperplane H crosses K and K(cid:48) if and only if H crosses G (K(cid:48)); the hyperplane H separates K,K(cid:48) if and only if H crosses γˆ. All remaining K hyperplanes either cross exactly one of K,K(cid:48) or fail to cross C. Observe that the set of hyperplanes 4 separatingK,K(cid:48) containsnotripleH,H(cid:48),H(cid:48)(cid:48) ofdisjointhyperplanes,noneofwhichseparatestheother two. (Suchaconfigurationiscalledafacingtriple.) 1.2.2. Salvetti complexes and special cube complexes. Let Γ be a simplicial graph and let A be the corre- Γ spondingright-angledArtingroup(RAAG),i.e. thegrouppresentedby (cid:10) (cid:11) V(Γ) | [v,w], {v,w} ∈ E(Γ) , whereV(Γ)andE(Γ)respectivelydenotethevertex-andedge-setsofΓ. ThephrasegeneratorofΓrefers tothispresentation;wedenoteeachgeneratorofA bythecorrespondingvertexofΓ. Γ The RAAG A is isomorphic to the fundamental group of the Salvetti complex S , introduced in [16], Γ Γ whichisanonpositively-curvedcubecomplexwithone0–cubex,anoriented1–cubeforeachv ∈ V(Γ), labeledbyv,andann–torus(ann–cubewithoppositefacesidentified)foreveryn–cliqueinΓ. AcubecomplexX isspecialifthereexistsasimplicialgraphΓandalocalisometryX → S inducing Γ a monomorphism π1X → AΓ and a π1X-equivariant embedding X(cid:101) → S(cid:101)Γ of universal covers whose image is a convex subcomplex. Specialness allows one to study geometric features of π X by working 1 inside of S(cid:101)Γ, which has useful structure not necessarily present in general CAT(0) cube complexes; see Section 1.2.3. Following Haglund-Wise [23], a group G is (virtually) [compact] special if G is (virtually) isomorphictothefundamentalgroupofa[compact]specialcubecomplex. 1.2.3. Cubical features particular to Salvetti complexes. Let Γ be a finite simplicial graph and let Λ be an induced subgraph of Γ. The inclusion Λ (cid:44)→ Γ induces a monomorphism A → A . In fact, there is an Λ Γ injective local isometry S → S inducing A → A . Hence each conjugate Ag of A in A is the Λ Γ Λ Γ Λ Λ Γ stabilizerofaconvexsubcomplexgS(cid:101)Λ ⊆ S(cid:101)Γ. Afewspecialcaseswarrantextraconsideration. When Λ ⊂ Γ is an n-clique, for some n ≥ 1, then S ⊆ S is an n–torus, which is the Salvetti Λ Γ complexofthesub-RAAGisomorphictoZn generatedbynpairwise-commutinggenerators. Inthiscase, SΛ is a standard n-torus in SΓ. (When n = 1, SΛ is a standard circle.) Each lift of S(cid:101)Λ to S(cid:101)Γ is a standard flat; when n = 1, we use the term standard line; a compact connected subcomplex of a standard line is a standard segment. The labels and orientations of 1–cubes in SΓ pull back to S(cid:101)Γ; a standard line is a convexsubcomplexisometrictoR, allofwhose1–cubeshavethesamelabel, suchthateach0–cubehas oneincomingandoneoutgoing1–cube. WhenLk(v)isthelinkofavertexv ofΓ,thesubcomplexS isanimmersedcombinatorialhyper- Lk(v) planeinthesensethatS(cid:101)Lk(v) isacombinatorialhyperplaneofS(cid:101)Γ. Thereisacorrespondinghyperplane, whose carrier is bounded by S(cid:101)Lk(v) and vS(cid:101)Lk(v), that intersects only 1–cubes labeled by v. Moreover, S(cid:101)Lkv is contained in S(cid:101)St(v), where St(v) is the star of v, i.e. the join of v and Lk(v). It follows that S(cid:101)St(v) ∼= S(cid:101)Lk(v) ×S(cid:101)v, whereS(cid:101)v isastandardline. NotethatthecombinatorialhyperplaneS(cid:101)Lk(v) ispar- alleltovkS(cid:101)Lk(v) forallk ∈ Z. Likewise,S(cid:101)v isparalleltogS(cid:101)v exactlywheng ∈ AΛ,andparallelstandard lines have the same labels. We say S(cid:101)v is a standard line dual to S(cid:101)Lk(v), and is a standard line dual to any hyperplaneH suchthatN(H)hasS(cid:101)Lk(v) asoneofitsboundingcombinatorialhyperplanes. Remark 1.3. We warn the reader that a given combinatorial hyperplane may correspond to distinct hyperplaneswhosedualstandardlineshavedifferentlabels;thisoccursexactlywhenthereexistmultiple vertices in Γ whose links are the same subgraph. However, the standard line dual to a genuine (non- combinatorial)hyperplaneisuniquely-determineduptoparallelism. Definition 1.4 (Frame). Let K ⊆ S(cid:101)Γ be a convex subcomplex and let H be a hyperplane. Let L be a standardlinedualtoH. TheframeofH istheconvexsubcomplexH(cid:48)×L ⊆ S(cid:101)Γ describedabove,where H(cid:48) isacombinatorialhyperplaneboundingN(H). IfK ⊆ S(cid:101)Γ isaconvexsubcomplex,andH intersects K,thentheframeofH inK isthecomplexK∩(H×L). Itisshownin[9]thattheframeofH inK has 5 theform(H ∩K)×(L∩K),providedthatLischoseninitsparallelismclasstointersectK. Notethat theframeofH isinfactwell-defined,sinceallpossiblechoicesofLareparallel. 2. ConsequencesofTheoremA Assuming Theorem A, we quantify separability of cubically convex-cocompact subgroups of special groupswiththeproofsofCorollariesBandC,beforeprovingTheoremAinthenextsection. ProofofCorollaryB. Let Γ be a finite simplicial graph so that there is a local isometry X → S . Let Γ Q ∈ Q berepresentedbyalocalisometryZ → X. Thenforallg ∈ π X −π Z, byTheoremA,there R 1 1 is a local isometry Y → S such that Y contains Z as a locally convex subcomplex, and g (cid:54)∈ π Y, and Γ 1 |Y(0)| ≤ |Z(0)|(|g|+1). Applyingcanonicalcompletion[23]toY → SΓyieldsacoverS(cid:98)Γ → SΓinwhich Y embeds;thiscoverhasdegree|Y(0)|by[9,Lemma2.8]. LetH(cid:48) = π1S(cid:98)Γ∩π1X,sothatπ1Z ≤ H(cid:48),and g (cid:54)∈ H(cid:48),and[π X : H(cid:48)] ≤ |Z(0)|(|g|+1). Thefirstclaimfollows. 1 LetG ∼= π1X withX compactspecial,Q ≤ G,andtheconvexhullofQx˜inX(cid:101) liesinNK(QX˜). Then the second claim follows since we can choose Z to be the quotient of the hull of Qx˜ by the action of Q, and|Z(0)| ≤ gr (K). (cid:3) X(cid:101) Ingeneral,thenumberof0–cubesinZ iscomputablefromthequasiconvexityconstantofaQ-orbitin X(cid:101)(1) by[22,Theorem2.28]). Inthehyperboliccase,weobtainCorollaryCintermsofthequasiconvexity constant,withoutreferencetoanyparticularcubecomplex: ProofofCorollaryC. We use Corollary B when J = 1, and promote the result to a polynomial bound whenJ ≥ 1. LetQ ∈ Q andletg ∈ G−Q. K The special case: Suppose J = 1 and let X be a compact special cube complex with G ∼= π X. Let 1 Z → X be a compact local isometry representing the inclusion Q → G. Such a complex exists by quasiconvexity of Q and [22, Theorem 2.28], although we shall use the slightly more computationally explicit proof in [36]. Let A(cid:48) ≥ 1,B(cid:48) ≥ 0 be constants such that an orbit map (G,(cid:107)−(cid:107)S) → (X(cid:101)(1),d) is an (A(cid:48),B(cid:48))-quasi-isometric embedding, where d is the graph-metric. Then there exist constants A,B, dependingonlyonA(cid:48),B(cid:48)andhenceon(cid:107)−(cid:107) ,suchthatQxis(AK+B)-quasiconvex,wherexisa0–cube S inZ(cid:101) ⊂ X(cid:101). BytheproofofProposition3.3of[36],theconvexhullZ(cid:101)ofQxliesintheρ-neighborhoodof √ (cid:16) (cid:16) (cid:17) (cid:17) Qx,whereρ = AK+B+ dimX+δ(cid:48) csc 1 sin−1(√ 1 +1 andδ(cid:48) = δ(cid:48)(δ,A(cid:48),B(cid:48)). CorollaryB 2 dimX provides G(cid:48) ≤ G with g (cid:54)∈ G(cid:48), and [G : G(cid:48)] ≤ |Z(0)|(|g|+1). But |g|+1 ≤ A(cid:48)(cid:107)g(cid:107) +B(cid:48) +1, while S |Z(0)| ≤ gr (ρ). Thus[G : G(cid:48)] ≤ gr (ρ)A(cid:48)(cid:107)g(cid:107) +gr (ρ)B(cid:48)+gr (ρ),sothatthereexistsP suchthat X(cid:101) X(cid:101) S X(cid:101) X(cid:101) 1 SepQK(Q,n) ≤ P gr (P K)n G,S 1 X(cid:101) 1 forallK,Q ∈ Q ,n ∈ N,whereP dependsonlyonX. K 1 The virtually special case: Now suppose that J ≥ 1. We have a compact special cube complex X, and [G : G(cid:48)] ≤ J!, where G(cid:48) ∼= π X and G(cid:48) (cid:47)G. Let Q ≤ G be a K-quasiconvex subgroup. By Lemma 2.1, 1 thereexistsC = C(G,S)suchthatQ∩G(cid:48) isCJ!(K+1)-quasiconvexinG,andthusisP CJ!(K+1)- 2 quasiconvexinG(cid:48),whereP dependsonlyonGandS. 2 Letg ∈ G−Q. SinceG(cid:48)(cid:47)G,theproductQG(cid:48) isasubgroupofGofindexatmostJ!thatcontainsQ. Hence, ifg (cid:54)∈ QG(cid:48), thenwearedone. Wethusassumeg ∈ QG(cid:48). Hencewecanchoosealefttransversal {q ,...,q } for Q∩G(cid:48) in Q, with s ≤ J! and q = 1. Write g = q g(cid:48) for some i ≤ s, with g(cid:48) ∈ G(cid:48). 1 s 1 i Suppose that we have chosen each q to minimize (cid:107)q (cid:107) among all elements of q (Q ∩ G(cid:48)), so that, by i i S i Lemma2.3,(cid:107)q (cid:107) ≤ J!foralli. Hence(cid:107)g(cid:48)(cid:107) ≤ ((cid:107)g(cid:107) +J!). i S S By the first part of the proof, there exists a constant P , depending only on G,G(cid:48),S, and a subgroup 1 G(cid:48)(cid:48) ≤ G(cid:48) suchthatQ∩G(cid:48) ≤ G(cid:48)(cid:48),andg(cid:48) (cid:54)∈ G(cid:48)(cid:48),and [G(cid:48) : G(cid:48)(cid:48)] ≤ P gr (P P CJ!(K +1))(cid:107)g(cid:48)(cid:107) ≤ P gr (P P CJ!(K +1))(cid:107)g(cid:48)(cid:107) . 1 G(cid:48) 1 2 S 1 G 1 2 S 6 LetG(cid:48)(cid:48)(cid:48) = ∩s q G(cid:48)(cid:48)q−1,sothatg(cid:48) (cid:54)∈ G(cid:48)(cid:48)(cid:48),andQ∩G(cid:48) ≤ G(cid:48)(cid:48)(cid:48) (sinceG(cid:48) isnormal),and i=1 i i [G(cid:48) : G(cid:48)(cid:48)(cid:48)] ≤ (P gr (P P CJ!(K +1))(cid:107)g(cid:48)(cid:107) )s. 1 G 1 2 S Finally, letH = QG(cid:48)(cid:48)(cid:48). This subgroup clearly contains Q. Suppose that g = q g(cid:48) ∈ H. Then g(cid:48) ∈ QG(cid:48)(cid:48)(cid:48), i i.e. g(cid:48) = ag(cid:48)(cid:48)(cid:48) forsomea ∈ Qandg(cid:48)(cid:48)(cid:48) ∈ G(cid:48)(cid:48)(cid:48). Sinceg(cid:48) ∈ G(cid:48) andG(cid:48)(cid:48)(cid:48) ≤ G(cid:48), wehavea ∈ Q∩G(cid:48), whence a ∈ G(cid:48)(cid:48)(cid:48),byconstruction. Thisimpliesthatg(cid:48) ∈ G(cid:48)(cid:48)(cid:48) ≤ G(cid:48)(cid:48),acontradiction. HenceH isasubgroupofG separatingg fromQ. Finally, [G : H] ≤ [G : G(cid:48)(cid:48)(cid:48)] ≤ J![P gr (P P CJ!(K +1))((cid:107)g(cid:107) +J!)]J!, 1 G 1 2 S andtheproofiscomplete. (cid:3) Lemma2.1. LetthegroupGbegeneratedbyafinitesetS andlet(G,(cid:107)−(cid:107) )beδ-hyperbolic. LetQ ≤ G S be K-quasiconvex, and let G(cid:48) ≤ G be an index-I subgroup. Then Q∩G(cid:48) is CI(K +1)-quasiconvex in G forsomeC dependingonlyonδ andS. Proof. Since Q is K-quasiconvex in G, it is generated by a set T of q ∈ Q with (cid:107)q(cid:107) ≤ 2K +1, by [13, S LemmaIII.Γ.3.5]. Astandardargumentshows(Q,(cid:107)−(cid:107) ) (cid:44)→ (G,(cid:107)−(cid:107) )isa(2K+1,0)-quasi-isometric T S embedding. Lemma2.3showsthatQ∩G(cid:48) isI-quasiconvexin(Q,(cid:107)−(cid:107) ),since[Q : Q∩G(cid:48)] ≤ I. Hence T Q∩G(cid:48)hasageneratingsetmakingit((2I+1)(2K+1),0)-quasi-isometricallyembeddedin(G,(cid:107)−(cid:107) ). S ApplyLemma2.2toconclude. (cid:3) Thefollowinglemmaisstandard,butweincludeittohighlighttheexactconstantsinvolved: Lemma2.2. LetGbeagroupgeneratedbyafinitesetS andsupposethat(G,(cid:107)−(cid:107) )isδ-hyperbolic. Then S there exists a (sub)linear function f : N → N, depending on S and δ, such that σ ⊆ N (γ) whenever f(λ) γ : [0,L] → Gisa(λ,0)–quasigeodesicandσ isageodesicjoiningγ(0)toγ(L). Proof. Seee.g. theproofof[13,TheoremIII.H.1.7]. (cid:3) Lemma 2.3. Let Q be a group generated by a finite set S and let Q(cid:48) ≤ Q be a subgroup with [Q : Q(cid:48)] = s < ∞. Thenthereexistsalefttransversal{q ,...,q }forQ(cid:48) suchthat(cid:107)q (cid:107) ≤ sfor1 ≤ i ≤ s. HenceQ(cid:48) 1 s i S iss-quasiconvexinQ. Proof. Supposethatq = s ···s isageodesicwordinS ∪S−1 andthatq isashortestrepresentative k ik i1 k of q Q(cid:48). Let q = s ···s be the word in Q consisting of the last j letters of q for all 1 < j < k, k j ij i1 k and let q = 1. We claim that each q is a shortest representative for q Q(cid:48). Otherwise, there exists p 1 j j with (cid:107)p(cid:107) < j such that q Q(cid:48) = pQ(cid:48). But then s ···s pQ(cid:48) = q Q(cid:48), and thus q was not a shortest S j k j+1 k k representative. It also follows immediately that q Q(cid:48) (cid:54)= q Q(cid:48) for j (cid:54)= j(cid:48). Thus, q ,q ,...,q represent j j(cid:48) 1 2 k distinctleftcosetsofQ(cid:48) providedk ≤ s,andtheclaimfollows. (cid:3) Remark2.4(Embeddingsinfinitecovers). GivenacompactspecialcubecomplexX andacompactlocal isometry Z → X, Theorem A gives an upper bound on the minimal degree of a finite cover in which Z embeds; indeed, producing such an embedding entails separating π Z from finitely many elements in 1 π X. However,itisobservedin[9,Lemma2.8]theHaglund–Wisecanonicalcompletionconstruction[23] 1 producesacoverX(cid:98) → X ofdegree|Z(0)|inwhichZ embeds. 3. ProofofTheoremA Inthissection,wegiveaproofofthemaintechnicalresult. Definition 3.1. Let SΓ be a Salvetti complex and let S(cid:101)Γ be its universal cover. The hyperplanes H,H(cid:48) of S(cid:101)Γ are collateral if they have a common dual standard line (equivalently, the same frame). Clearly collateralism is an equivalence relation, and collateral hyperplanes are isomorphic and have the same stabilizer. 7 Being collateral implies that the combinatorial hyperplanes bounding the carrier of H are parallel to those bounding the carrier of H(cid:48). However, the converse is not true when Γ contains multiple vertices whose links coincide. In the proof of Theorem A, we will always work with hyperplanes, rather than combinatorialhyperplanes,unlessweexplicitlystatethatwearereferringtocombinatorialhyperplanes. ProofofTheoremA. Let x˜ ∈ S(cid:101)Γ be a lift of the base 0–cube x in SΓ, and let Z(cid:101) ⊆ S(cid:101)Γ be the lift of the universal cover of Z containing x˜. Since Z → SΓ is a local isometry, Z(cid:101) is convex. Let Z(cid:98) ⊂ Z(cid:101) be the convex hull of a compact connected fundamental domain for the action of π1Z ≤ AΓ on Z(cid:101). Denote by K theconvexhullofZ(cid:98)∪{gx˜}andletSbethesetofhyperplanesofS(cid:101)Γ intersectingK. Wewillforma quotientofK,restrictingtoZ(cid:98) → Z onZ(cid:98),whoseimageadmitsalocalisometrytoSΓ. Thesubcomplex(cid:98)Z(cid:98)(cid:99): LetLbethecollectionofstandardsegments(cid:96)inK thatmaptostandardcircles inSΓwiththepropertythat(cid:96)∩Z(cid:98)hasnon-contractibleimageinZ. Let(cid:98)Z(cid:98)(cid:99)beconvexhullofZ(cid:98)∪(cid:83)(cid:96)∈L(cid:96), sothatZ(cid:98) ⊆ (cid:98)Z(cid:98)(cid:99) ⊆ K. PartitioningS: WenowpartitionSaccordingtothevarioustypesofframesinK. First,letZbetheset ofhyperplanesintersectingZ(cid:98). Second,letNbethesetofN ∈ S−Zsuchthattheframe(N∩K)×(L∩K) of N in K has the property that for some choice of x0 ∈ N(0), the segment ({x0}×L)∩Z(cid:98) maps to a nontrivialcycleof1–cubesinZ. LetnN ≥ 1bethelengthofthatcycle. ByconvexityofZ(cid:98), thenumber n is independent of the choice of the segment L within its parallelism class. Note that N is the set of N hyperplanes that cross (cid:98)Z(cid:98)(cid:99), but do not cross Z(cid:98). Hence each N ∈ N is collateral to some W ∈ Z. Third, fixacollection{H ,...H } ⊂ S−Zsuchthat: 1 k (1) For1 ≤ i ≤ k−1,thehyperplaneHi separatesHi+1 from(cid:98)Z(cid:98)(cid:99). (2) For1 ≤ i < j ≤ k, ifahyperplaneH separatesH fromH , thenH iscollateraltoH forsome i j (cid:96) (cid:96) ∈ [i,j]. Similarly,ifH separatesH1 from(cid:98)Z(cid:98)(cid:99),thenH iscollateraltoH1,andifH separatesHk fromgx˜,thenH iscollateraltoH . k (3) Foreachi,theframe(H ∩K)×L ofH inK hasthepropertythatforeveryh ∈ H(0),theimage i i i i inZ ofthesegment({h}×Li)∩Z(cid:98)isemptyorcontractible. (Here,Li isastandardsegmentofa standardlinedualtoH .) i Let H be the set of all hyperplanes of S−Z that are collateral to H for some i. Condition (3) above i ensuresthatH∩N = ∅,whileH = ∅onlyifK = (cid:98)Z(cid:98)(cid:99). Finally,letB = S−(Z∪N∪H). Notethateach B ∈ BcrossessomeH . Figure1showsapossibleK andvariousfamiliesofhyperplanescrossingit. i N2 B2 H2 H3 N1 B1 H1 Figure1. HyperplanescrossingK (thedarkshadedareaontheleftisZ(cid:98)). 8 Mapping (cid:98)Z(cid:98)(cid:99) to Z: We now define a quotient map q : (cid:98)Z(cid:98)(cid:99) → Z extending the restriction Z(cid:98) → Z of Z(cid:101) → Z. NotethatifN = ∅,then(cid:98)Z(cid:98)(cid:99) = Z(cid:98),andqisjustthemapZ(cid:98) → Z. HencesupposeN (cid:54)= ∅andlet N ,...,N bethecollateralismclassesofhyperplanesinN,andfor1 ≤ i ≤ s,letN(cid:48) bethecollateralism 1 s i class of N in S, i.e. N together with a nonempty set of collateral hyperplanes in Z. For each i, let L i i i be a maximal standard line segment of (cid:98)Z(cid:98)(cid:99), each of whose 1–cubes is dual to a hyperplane in N(cid:48) and i which crosses each element of N(cid:48)i. For each i, let Ni ∈ Ni be a hyperplane separating Z(cid:98) from gx˜. Then Ni∩Nj (cid:54)= ∅fori (cid:54)= j,sinceneitherseparatestheotherfromZ(cid:98). WecanchoosetheLi sothatthereisan isometricembedding(cid:81)ki=1Li → (cid:98)Z(cid:98)(cid:99),sincewhetherornottwohyperplanesofS(cid:101)Γcrossdependsonlyon theircollateralismclasses. For each nonempty I ⊆ {1,...,k}, a hyperplane W ∈ Z crosses some U ∈ ∪ N(cid:48) if and only i∈I i if W crosses each hyperplane collateral to U. Hence, by [19, Lemma 7.11], there is a maximal convex subcomplex Y(I) ⊂ Z(cid:98), defined up to parallelism, such that a hyperplane W crosses each U ∈ ∪i∈IN(cid:48)i if and only if W ∩ Y(I) (cid:54)= ∅. Let A(I) be the set of hyperplanes crossing Y(I). By the definition of Y(I)and[19,Lemma7.11],thereisacombinatorialisometricembeddingY(I)×(cid:81)i∈ILi → (cid:98)Z(cid:98)(cid:99),whose image we denote by F(I) and refer to as a generalized frame. Moreover, for any 0–cube z ∈ (cid:98)Z(cid:98)(cid:99) that is notseparatedfromahyperplanein∪ N(cid:48)∪A(I)byahyperplanenotinthatset,wecanchooseF(I)to i∈I i containz (thisfollowsfromtheproofof[19, Lemma7.11]). Figures2, 3, 4, and5showpossibleN(cid:48)’sand i generalizedhyperplaneframes. N(cid:48) 1 N(cid:48) 2 Figure 2. Collateral families N(cid:48) andN(cid:48). Figure3. Y({1})×L . 1 2 1 Figure4. Y({2})×L Figure5. Y({1,2})×(L ×L ) 2 1 2 Tobuildq,wewillexpress(cid:98)Z(cid:98)(cid:99)astheunionofZ(cid:98)andacollectionofgeneralizedframes,defineqoneach generalizedframe,andcheckthatthedefinitioniscompatiblewheremultiplegeneralizedframesintersect. Let z ∈ (cid:98)Z(cid:98)(cid:99) be a 0–cube. Either z ∈ Z(cid:98), or there is a nonempty set I ⊂ {1,...,k} such that the set of hyperplanesseparatingz fromZ(cid:98)iscontainedin∪i∈IN(cid:48)i,andeachN(cid:48)i containsahyperplaneseparatingz 9 fromZ(cid:98). IfH ∈ A(I)∪(cid:83) N(cid:48) isseparatedfromzbyahyperplaneU,thenU ∈ A(I)∪(cid:83) N(cid:48),whence i∈I i i∈I i wecanchooseF(I)tocontainz. Hence(cid:98)Z(cid:98)(cid:99)istheunionofZ(cid:98)andafinitecollectionofgeneralizedframes F(I ),...,F(I ). 1 t Foranyp ∈ {1,...,t},wehaveF(I ) = Y(I )×(cid:81) L andweletY(I ) = im(Y(I ) → Z)and p p j∈Ip j p p let Lj = im(Lj ∩Z(cid:98) → Z) be the cycle of length nNj to which Lj maps, for each j ∈ Ip. Note that Z containsF(I ) = Y(I )×(cid:81) L andsowedefinethequotientmapq : F(I ) → Z astheproductof p p j∈IP j p p theabovecombinatorialquotientmaps. Namely,q (y,(r ) ) = (y,(r mod n ) )fory ∈ Y(I ) p j j∈IP j Nj j∈Ip p andr ∈ L . j j Toensurethatq (F(I )∩F(I )) = q (F(I )∩F(I ))foralli,j ≤ t,itsufficestoshowthat p p j j p j     (cid:89) (cid:89) (cid:89) F(Ip)∩F(Ij) := Y(Ip)× Lk∩Y(Ij)× L(cid:96) = [Y(Ip)∩Y(Ij)]× Lk. k∈Ip (cid:96)∈Ij k∈Ip∩Ij Thisinturnfollowsfrom[15,Proposition2.5]. Hence,thequotientmapsq arecompatibleandtherefore p defineacombinatorialquotientmapq : (cid:98)Z(cid:98)(cid:99) → Z extendingthemapsqp. Observe that if H = ∅, i.e. K = (cid:98)Z(cid:98)(cid:99), then we take Y = Z. By hypothesis, Z admits a local isometry toS andhasthedesiredcardinality. Moreover, ourhypothesisong ensuresthatg (cid:54)∈ π Y, butthemap Γ 1 q shows that any closed combinatorial path in S representing g lifts to a (non-closed) path in Z, so the Γ proofofthetheoremiscomplete. ThuswecanandshallassumethatH (cid:54)= ∅. Quotients of H-frames: To extend q to the rest of K, we now describe quotient maps, compatible with themapZ(cid:98) → Z,onframesassociatedtohyperplanesinH. AnisolatedH-frameisaframe(H ∩K)×L, whereH ∈ HandH crossesnohyperplaneofZ(cid:98)(andhencecrossesnohyperplaneof(cid:98)Z(cid:98)(cid:99)). Aninterfered H-frameisaframe(H∩K)×L,whereH ∈ HandH crossesanelementofZ. Equivalently,(H∩K)×L isinterferedifg (Z(cid:98))containsa1–cubeandisisolatedotherwise. N(H) DefinequotientmapsonisolatedH-framesbythesamemeansaswasusedforarbitraryframesin[9]: let(H ∩K)×LbeanisolatedH-frame. LetH betheimmersedhyperplaneinS towhichH issentby Γ S(cid:101)Γ → SΓ,andletH ∩K betheimageofH∩K. WeformaquotientYH = H ∩K×Lofeveryisolated H-frame(H ∩K)×L. NowwedefinethequotientsofinterferedH-frames. LetA(cid:98)= g (Z(cid:98))andletAbeimageofA(cid:98)under N(H) Z(cid:98) → Z.... There is a local isometry A → SΓ, to which we app.l.y. canonical c(cid:12)o.m.. pl(cid:12)etion to produce a finite coverSΓ → SΓ whereAembeds. By Lemma 2.8 of [9], deg(SΓ → SΓ) = (cid:12)(cid:12)S(Γ0)(cid:12)(cid:12) = (cid:12)(cid:12)A(0)(cid:12)(cid:12) ≤ (cid:12)(cid:12)Z(0)(cid:12)(cid:12). Let ... H ∩K = im(H ∩K → S ),andmaptheinterferedH-frame(H ∩K)×LtoY = H ∩K ×L. Γ H Constructing Y: We now construct a compact cube complex Y(cid:48) from Z and the various quotients Y . H AhyperplaneW inK separatesH1fromZ(cid:98)onlyifW ∈ N. EachH-hyperplaneframehastheform(Hi∩ K)×L = (H ∩K)×[0, m ],parametrizedsothat(H ∩K)×{0}istheclosestcombinatorialhyperplane i i i i in the frame to Z(cid:98). We form Y(cid:48)(1) by gluing YH1 to Z along the image of gZ(cid:98)((H1∩K)×{0}), enabled by the fact that the quotients of interfered H-frames are compatible with Z(cid:98) → Z. In a similar manner, formY(cid:48)(i)fromY(cid:48)(i−1)andY byidentifyingtheimageof(H ∩K)×{m }∩(H ∩K)×{0} Hi i−1 i−1 i inY ⊂ Y(cid:48)(i−1)withitsimageinY . LetY(cid:48) = Y(cid:48)(k). Hi−1 Hi LetK(cid:48) = (cid:98)Z(cid:98)(cid:99)∪(cid:2)(cid:83)Hi∈H(Hi∩K)×Li(cid:3). Since Hi ∩Hj = ∅fori (cid:54)= j, there exists amap (K(cid:48),x˜) → (Y(cid:48),x) and a map (Y(cid:48),x) → (S ,x) such that the composition is precisely the restriction to K(cid:48) of the Γ coveringmap(S(cid:101)Γ,x˜) → (SΓ,x). IfY(cid:48) → S failstobealocalisometry,thenthereexistsiandnontrivialopencubese ⊂ H ∩K × Γ i−1 {m } (or Z if i = 1) and c ⊂ H ∩K ×{0} such that S contains an open cube e¯×c¯, where e¯,c¯are i−1 i Γ the images of e,c under S(cid:101)Γ → SΓ, respectively. Moreover, since g (Hi∩K) ⊆ g (Hi−1∩K), we can Z(cid:98) Z(cid:98) 10