Coarse medians and universal quasigeodesic cones

Coarse medians and univ ersal quasigeodesic cones Robert T ang 16 Mar ch 2026 Abstract W e sho w that any universal quasigeodesic cone of uniformly coarse median spaces admits a canonical coarse median structur e. As an application, we r ecover a r esult of Bo wditch which states that any hierarchically hyperbolic space admits a coarse me- dian structure compatible with the projections to the hyperbolic factor spaces . MSC2020 Classification : 51F30, 20F65 Keywor ds : coarse median, universal quasigeodesic cone , hierar chically hyperbolic space 1 Int roduction Coarse median spaces and groups , introduced b y Bo wditch [ Bow13a ], are eq uipped with a ternar y operator which captures useful features of nonpositive curvature. These gen- eralise both Gromo v hyperbolic spaces and CA T(0)–cube complexes, and include map- ping class groups, T eichmüller space, and geometrically finite Kleinian groups among ex- amples [ Bo w16 ]. The coarse median viewpoint has been productive, pro viding a unified framework for questions about Dehn functions, asymptotic cones [ Bo w13a ], quasi-isometric rigidity [ Bo w18 ], automorphisms [ Fio24 ], and analytic properties [ Zei16 , SW17 ]. A natural coarse median on the mapping class groups arises from the Behrstock–M insky centroid construction: subsurface projections to cur ve graphs yield centroids in each hy- perbolic factor [ MM99 , MM00 ]; these then assemble to a global centroid whose projec- tions coarsely agree with the factor wise centroids [ BM11 ]. I n [ Bo w18 ], Bo wditch extends this approach to hierarchically hyperbolic spaces (HHS), as developed by Behrstock–H agen– Sisto [ BHS17a , BHS19 ], showing that coarse medians on unifor mly hyperbolic factor spaces induce a coarse median on the total space uniformly compatible with the projection data. In this note, we revisit these “ pulling back ” constructions from the perspective of universal quasigeodesic cones, as introduced in a recent paper of the author [ T an ]. I nformally , these act as universal quasigeodesic receivers over diagrams of metr ic spaces equipped with uniformly controlled maps between them (see Section 2 for the precise definition). W e pro ve that if the factor spaces admit uniform coarse medians so that the given maps ar e uniformly coarse median preserving, then they induce a compatible coarse median on the universal quasigeodesic cone; see Theorem 4.4 for a pr ecise statement. Theorem 1.1. Let D be a uniformly controlled diagram of uniformly coarse median spaces p X i , µ i q whose maps are uniformly coarse median preserving. Then any universal quasi- geodesic uniformly controlled cone X o ver D admits a canonical (up to closeness) coarse median µ such that all legs p X , µ q Ñ p X i , µ i q ar e uniformly coarse median p r eserving. 2 Robert T ang One main r esult from [ T an ] is that any HHS realises a universal quasigeodesic cone o ver its diagram of hyperbolic factor spaces together with the usual pair wise constr aint spaces. Theorem 1.1 applies to such diagrams and so we reco ver Bo wditch ’ s coarse median theo- rem for HHSs (see Section 5 ). Mor e generally , this applies to families of uniformly coarse median spaces equipped with pair wise constraints either imposing no constraint or a bounded coarse interval image property . Theorem 1.1 further gives uniqueness of the coarse median on an HHS (up to closeness) compatible with given HHS–data; we remark that without prescribing HHS–data an HHS may admit (possibly u ncountably) many dis- tinct coarse medians [ Man ], though uniqueness does hold in some settings [ FS ]. The proof of Theorem 1.1 is constru ctive: b y our earlie r work [ T an ], a universal quasi- geodesic cone can be realised using an explicit Rips–tuple space associated to the data; the desired coarse median is then induced by the product median. This highlights the utility of these tools for constructively promoting factorwise structures to global ones. Ackno wledgements The author thanks Mark Hagen, Robert Krop holler , H arr y P etyt, Alessandro Sisto , Davide Spriano , and Jing T ao for interesting conversations. The author acknowledges support from the N ational N atural Science F oundation of China (NSFC 12101503); the Suzhou Sci- ence and T echnology Development Planning Programme (ZXL2022473); and the XJTLU Resear ch Development Fund (RDF-23-01-121). 2 Coarse geometry Let X and Y be (extended) metr ic spaces. An upper control for a map f : X Ñ Y is a proper increasing function ρ : r 0, 8q Ñ r 0, 8q such that such that d X p x , x 1 q ď t implies d Y p f x , f x 1 q ď ρ p t q for all x , x 1 P X and t ě 0. W e say f is controlled if it admits an u pper control, and coarsely Lipschitz if it admits an affine upper control. For any subset U Ď Y , let N r p U q denote its closed metric r –neighbourhood in Y . W e call f coarsely surjective if N r f p X q “ Y for some r ě 0. For κ ě 0 and points y , y 1 P Y , wr ite y « κ y 1 to mean d Y p y , y 1 q ď κ . Say two maps f , g : X Ñ Y are κ –close , denoted as f « κ g , if f p x q « κ g p x q for all x P X . W e say f , g ar e close , and denote this by f « g , if f « κ g for some κ ě 0. N ote that closeness depends only on the codomain metric. A controlled map f : X Ñ Y is a coarse equivalence if there exists a controlled map g : Y Ñ X such that g f « 1 X and f g « 1 Y ; such a map g is called a coarse inverse for f , and is unique up to closeness. If, in addition, f and g are coarsely Lipschitz then we call f a quasi-isometry . 2.1 U niformly controlled cones W e recall the notions of uniformly controlled diagrams and cones from [ T an ]. Let Coarse be the category with extended metric spaces asobjects, and controlled maps as morphisms. Let J be a directed multigraph, with vertex set Obj p J q and edge set M or p J q . A uniformly controlled (u.c.) diagr am D : J ù Coarse (of shape J ) comprises: • a metric space D j for each j P Obj p J q , and • a controlled map D φ : D i Ñ D j for each arro w φ : i Ñ j in M or p J q , such that all D φ admit a common upper control. Coarse medians and univ ersal quasigeodesic cones 3 The “ squiggly arro w” notation i s intended to suggest an analogy with diagrams in the usual categorical sense, but with uniform control and error involved. W e do not requir e exis- tence of compositions or coarse commutativity for u.c. diagrams . Let W be a metr ic space. A uniformly controlled cone λ : W D ov er D with apex W comprises a uniformly controlled family of legs λ j : W Ñ D j for j P Obj p J q such that W D i D j Ð Ñ λ i Ð Ñ λ j Ð Ñ D φ uniformly coarsely commutes for all arrows φ : i Ñ j . In other wor ds, there exists κ ě 0 such that λ j « κ D φ λ i for all arrows φ . T wo u.c. cones λ , λ 1 : W D are close , denoted as λ « λ 1 , if their legs are uniformly close . W e call λ : W D a u.c. limit cone o ver D if for any u.c. cone ζ : Z D , there exists a unique (up to closeness) controlled map f : Z Ñ W such that ζ « λ f . In other words , the diagram Z W D j Ð Ñ f Ð Ñ ζ j Ð Ñ λ j uniformly coarsel y commutes for all legs. Any u.c. limit cone o ver D , if it exists, is unique up to coarse equivalence (unique up to closeness); we shall denote it b y lim ø J D . F or κ ě 0, the κ –consistent tuplespace of D is T uple κ p D q : “ # p x j q j P ź j D j : x j « κ D φ x j for all arro ws φ + Ď ź j D j equipped with the induced ℓ 8 –metric. The 1–Lipschitz projections π κ j : T uple κ D Ñ D j form the legs of a u.c. cone π κ : T uple κ p D q D . The follo wing is immediate. Lemma 2.1. Let ζ : Z D be a u.c. cone. Then for all κ large, ζ factors as ζ “ π κ f κ wher e f κ : Z Ñ T uple κ D Ď ś j D j is induced by ś j ζ j . The inclusions T uple κ p D q ã Ñ T uple κ 1 p D q are isometr ic embedding for all κ ď κ 1 ; th ese give rise to a filtration T uple ˚ p D q . Say that T uple ˚ p D q stabilises if these inclusions are coarsely surjective for all κ ď κ 1 large . Proposition 2.2 (Stable tuplespaces [ T an , Proposition 5.17]) . Let D : J ù Coarse be a u.c. diagram. Then D admit s a u.c. limit cone if and only i f T uple ˚ p D q stabilises. In that case, π κ : T uple κ p D q D realises the u.c. limit cone o ver D for κ large. 2.2 U niversal quasigeodesic cones W e no w tur n our attention to quasigeodesic u.c. cones, that is, where the apex is quasi- geodesic. Recall that a metr ic space is quasigeodesic (r esp . coarsel y geodesic ) if it is quasi- isometric (resp . coarsely equivalent) to a graph with the standard graph (i.e. combina- torial) metri c; this is equivalent to the usual definition in terms of quasi- (resp . coarse) 4 Robert T ang geodesics [ T an , Proposition 3.20]. Any controlled map with quasigeodesic domain is nec- essarily coarsely Lipschitz. In particular , the legs of any quasigeodesic u.c. cone are uni- formly coarsely Lipschitz. Say that λ : W D is a universal quasigeodesic u.c. cone if every quasigeodesic u.c. cone ζ : Z D factors (up to closeness) through λ via a unique (up to closeness) coarsely Lipschitz map . Any such cone is unique up to quasi-isometr y . T o compute universal quasigeodesic u.c. cones, we employ the Rips filtration. Given a metric space X , the Rips gr aph Rips σ X at scale σ ě 0 is the graph with ver tex set X , with points x , x 1 P X declared adjacent if and only if d X p x , x 1 q ď σ . W e shall r egard Rips σ X as a metric space with underlying set X , where the metric is the restriction of the standar d graph metric. The underlying identity on X induces a • Lipschitz map ξ σ : Rips σ X Ñ X for all σ ě 0, and • a 1–Lipschitz map Rips σ X Ñ Rips τ X for all σ ď τ . By consider ing the Rips graph at all scales, we obtain the (metric) Rips filtration Rips ˚ X . Say that Rips ˚ X stabilises if the maps Rips σ X Ñ Rips τ X are coarse equivalences (or , equivalently , quasi-isometries) for all σ ď τ large. The next two results are stan dard. Lemma 2.3 ([ T an , Lemma 3.22]) . A metric space X is coarsely geodesic if and only if Rips ˚ X stabilises; in which case ξ σ : Rips σ X Ñ X is a coarse equivalence for σ large. Proposition 2.4 ([ T an , Lemma 3.3 and Proposition 3 .20]) . Let X be a quasigeodesic space and f : X Ñ Y be a controlled map. Then for σ ě 0 large, the map f σ : X Ñ Rips σ Y in- duced by f is controlled, hence coarsely Lipschitz. By applying the Rips filtr ation to the coarse tuplespaces, we obtain an explicit method for computing universal quasigeodesic u.c. cones . Theorem 2.5 (Rips–tuple r ecipe [ T an , Theorem 5.19]) . Let D : J ù Coa rse be a u.c. dia- gram. The following ar e equivalent. 1. D admits a universal quasigeodesic u.c. cone, 2. π κ ξ σ : Rips σ T uple κ D D is a un iversal quasigeodesic cone for some σ , κ ě 0 , 3. T uple ˚ D stabili ses to a coarsely geodesic space. 2.3 Coarse median spaces Background on coarse medians spaces can be found i n [ Bo w13a , Bow13b , Bow19 ]. W e shall use an equivalent formulation due to Niblo–W r ight–Zhang [ NWZ19 ]. Definition 2.6. A controlled map µ : X 3 Ñ X is a coarse median on a metric space X if there exists C ě 0 such that 1. (Coarse symmetry) µ p x τ 1 , x τ 2 , x τ 3 q « C µ p x 1 , x 2 , x 3 q for all permutations τ P S 3 , 2. (Coarse localisation) µ p x 1 , x 1 , x 2 q « C x 1 3. (Coarse four-point condition) µ p µ p x 1 , w , x 2 q , w , x 3 q « C µ p x 1 , w , µ p x 2 , w , x 3 qq for all x 1 , x 2 , x 3 , w P X . Call p X , µ q a coarse median space with coarse median con stant C . Coarse medians and univ ersal quasigeodesic cones 5 H ere , we work with the ℓ 8 –metric on X 3 . Any function µ 1 : X 3 Ñ X close to a coarse median µ on X is also a coarse median. W e say two coarse medians µ , µ 1 on X ar e close if they are close as controlled maps µ , µ 1 : X 3 Ñ X . Let r µ s denote the closeness class of a coarse median µ . A controlled map f : p X , µ q Ñ p Y , ν q between coarse median spaces is coarse-median pr e- serving (CMP) if f µ « ν ˝ f ˆ 3 : X 3 Ñ Y . In other words , there exists κ ě 0 such that f p µ p x , y , z qq « κ ν p f x , f y , f z q for all x , y , z P X . The assumption that the involved maps are contr olled ensures that the CMP pr operty is preserved under composition. Lemma 2.7. L et p X , µ q be a coarse median space and suppose that f : W Ñ X is a coarse equivalence. Then ther e exists a unique (up to closeness) coarse median ν on W such that f : p W , ν q Ñ p X , µ q is CMP . Proof. Let g : X Ñ W be a coarse inverse of f . It is straightforward to verify that the ternar y operator ν : “ f ˝ µ ˝ g ˆ 3 : Y 3 Ñ Y defines a coarse median on Y . N ote that ν ˝ f ˆ 3 « f ˝ µ ˝ g ˆ 3 ˝ f ˆ 3 « f µ ˝ 1 ˆ 3 W « f µ , hence f : p W , ν q Ñ p X , µ q is CMP . T o verify uniqueness, suppose ν 1 is coarse median on Y such that f : p W , ν 1 q Ñ p X , µ q is CMP . Then ν « ν ˝ f ˆ 3 ˝ g ˆ 3 « ν 1 ˝ f ˆ 3 ˝ g ˆ 3 « ν 1 as desired. Let p X , µ q be a coarse median space with coarse median constant C ě 0 and upper control ρ for µ . G iven points x , y P X and L ě 0, define their L –coarse interval to be r x , y s L : “ t z P X : µ p x , y , z q « L z u . Lemma 2.8. Let x , y P X . Then 1. x , y P r x , y s 2 C and µ p x , y , z q P r x , y s C ` ρ C for all z P X , 2. N r r x , y s L Ď r x , y s L ` r ` ρ r for all L , r ě 0 , and 3. If z P r x , y s L then r x , z s L Ď r x , y s L 1 wher e L 1 depends only on C , L , and ρ . Proof. 1. Coarse symmetry and coarse localisation imply that x , y P r x , y s 2 C , while µ p x , y , µ p x , y , z qq « C µ p µ p x , y , x q , y , z q « ρ C µ p x , y , z q follo ws from coarse symmetry and the coarse 4–point condition. 2. Let w « r z P r x , y s L . Then µ p x , y , w q « ρ r µ p x , y , z q « L z « r w . 3. Let w P r x , z s L . Then w « L ` C µ p w , x , z q « ρ L µ p w , x , µ p x , y , z qq « ρ C µ p w , x , µ p z , x , y qq « C µ p µ p w , x , z q , x , y q « ρ p C ` L q µ p w , x , y q « C µ p x , y , w q . 6 Robert T ang N ext, we recall a coarse version of the 5–point condition from median algebras . This fol- lo ws from Bo wditch ’ s original definition of coarse medians; see [ NWZ21 , Section 2.3]. Lemma 2.9. There exists a constant E ě 0 depending on C and ρ such that µ p v , w , µ p x , y , z qq « E µ p µ p v , w , x q , µ p v , w , y q , z q for all v , w , x , y , z P X . Lemma 2.10. If O P r x , y s L X r y , z s L X r z , x s L then O « R µ p x , y , z q where R depends on ρ , L , and C . Proof. Let m “ µ p x , y , z q . By the coarse 5–point condition, we deduce µ p O , m , x q « ρ L µ p µ p x , y , O q , µ p x , y , z q , x q « E µ p x , y , µ p O , z , x qq « ρ p L ` C q µ p x , y , O q « L O . Choosing K 1 ě ρ L ` E ` ρ p L ` C q ` L , it follo ws that m « C µ p m , m , O q « ρ p C ` C 1 q µ p µ p y , z , x q , µ p y , z , m q , O q « E µ p y , z , µ p x , m , O qq « ρ p K 1 ` C q µ p y , z , O q « L O . F ollowing Fioravanti [ Fio24 ], a subset U Ď X is an approximate median subalgebr a if there exists R ě 0 such that µ p U 3 q Ď N R p U q . F or such a subset, define a ternar y operator µ | U on U by choosing µ | U p x , y , z q P U to be any point within distance R of µ p x , y , z q in X . U p to closeness, µ | U is the unique coarse median on U which is close to the restriction of µ to U 3 Ď X 3 ; in particular , r µ | U s depends only on r µ s . W e say that µ | U is induced by µ . Any subset U 1 Ď X at finite Hausdorff distance from U is also an approximate median subalgebra. 3 Coarse median cones W e now consider u.c. diagrams and cones in the coarse median setting. The coarse median category CMed has coarse median spaces as objects and CMP controlled maps as mor- phisms. A u.c. coarse median diagram M : J ù CMed is a u.c. diagram with uniformly coarse median spaces M j : “ p X j , µ j q as objects (i.e., they admit a common coarse median constant and upper control), and whose bonding maps M φ : p X i , µ i q Ñ p X j , µ j q ar e uni- formly CMP . Let ˆ M : J ù Coa rse denote the underlying u.c. diagram of metric spaces, obtained by forgetting the coarse medians. A u.c. coarse median cone λ : p W , ν q M is a u.c. cone ov er ˆ M with a coarse median apex p W , ν q whose legs ar e uniformly CMP . C all λ a universal u.c. coarse median cone if ever y u.c. coarse median cone ζ : p Z , η q M factors through λ via a unique (up to closeness) CMP controlled map . Fix a common coarse median constant C ě 0 and upper control ρ for the objects p M j , µ j q , and c ě 0 such that µ j ˝ p M φ q ˆ 3 « c p M φ q ˝ µ i for all arro ws φ : i Ñ j . The uniformity assumptions allow us to equip the ℓ 8 –product ś j X j with the product coarse median ś j µ j ; this coarse median space shall be denoted as ś j M j . Say that a subset U Ď ś j M j is M –compatible if U Ă T uple κ ˆ M for some κ ě 0. F or such a set U , we may define a u.c. cone π | U : U ˆ M where the legs p π | U q j are restrictions of Coarse medians and univ ersal quasigeodesic cones 7 the projection to each M j –factor . If, in addition, U is an approximate median subalgebra of ś j M j then π | U : p U , p ś j µ j q| U q M is a u.c. coarse median cone. (Indeed, each factorwise projection π i : ś j M j Ñ M i is uniformly CMP .) N ote that if U is M –compatible (resp . an appro ximate median subalgebra) then so is N r p U q for any r ě 0. W e show that any u.c. coarse median cone λ : p W , ν q M factors canonically (up to close- ness) through an M –compatible appro ximate median subalgebra. Let I p λ q Ď ś j M j de- note the image of ś j λ j : p W , ν q Ñ ś j M j . Lemma 3.1. Let λ : p W , ν q M be a u.c. coarse median cone. Then ś j λ j is CMP and I p λ q is an M –compatible approximate median subalgebra. Proof. By assumption, ther e exists r ě 0 such that λ j ν « r µ j λ ˆ 3 j for all objects j , and so p ź j λ j q ˝ ν « r p ź j µ j q ˝ p ź j λ j q ˆ 3 . N ote that I p λ q 3 “ p ś j λ j q ˆ 3 p W 3 q . Thus, p ś j µ j qp I p λ q 3 q Ď N r p I p λ qq , hence I p λ q is an appro ximate median subalgebra. B y Lemma 2.1 , I p λ q is M –compatible. Therefor e λ “ π | I p λ q ˝ p ś j λ j q yields the desired factorisation, upon equipping I p λ q with the coarse median induced by ś j µ j . Thus, to characterise universal u.c. coarse median cones over M , we may focus our attention on M –compatible approximate median subal- gebras of ś j M j . Define an or der ing on subsets U , U 1 Ď ś j M j b y declaring U ă U 1 if and only if U Ď N r p U 1 q for some r ě 0. Proposition 3.2. A u. c. coarse median diagram M : J ù CMed admits a universal u.c. coarse median cone if and only if ther e exists a ă –greatest M –compatible appro ximate median subalgebra U of ś j M j . In that case, a universal u.c. coarse median cone is r ealised by π | U . Proof. Assume λ : p W , ν q M is a universal u.c. coarse median cone. By Lemma 3.1 , I p λ q is an M –compatible appro ximate median subalgebra of ś j M j . Let U Ď ś j M j be any M –compatible approximate median subalgebra. Then π | U : p U , p ś j µ j q| U q M is a u.c. coarse median cone . By the universal pr operty , π | U factors through λ up to closeness , and so I p π | U q “ U lies in a bounded neighbourhood of I p λ q , hence U ă I p λ q . F or the converse, assume U Ď ś j M j is ă –great est among M –compatible approximate median subalgebras. F or any u.c. coarse median cone ζ : p Z , ν q M , the set I p ζ q is an M –compatible approximate median subalgebra, by Lemma 3.1 , hence I p ζ q Ď N r p U q for some r ě 0. Let h be a coarse inverse to the inclusion U ã Ñ N r p U q . N ote that h is CMP with respect to the induced coarse medians . Then ζ factors as ζ “ π | N r p U q ˝ ź j ζ j « π | U ˝ h ˝ ź j ζ j . T o verify uniqueness (up to closeness), suppose that π | U ˝ h 1 « s π | U ˝ h 2 for some CMP controlled maps h 1 , h 2 : p Z , ν q Ñ p U , p ś j µ j q| U q and s ě 0. Then h 1 z “ p π j h 1 z q j “ pp π | U q j h 1 z q j « s pp π | U q j h 2 z q j “ p π j h 2 z q j “ h 2 z for all z P Z , hence h 1 « h 2 . Therefor e, π | U is a universal u.c. coarse median cone . 8 Robert T ang W e no w show that M induces a coarse median on its u.c. limit cone (if it exists). Theorem 3.3. Let M : J ù CMed be a u.c. coarse median diagram such that lim ø J ˆ M exists. Then T uple κ ˆ M is a ă –gr eatest M –compatible approximate median subalgebra of ś j M j for κ large. Consequently , a universal u.c. coarse median cone over M is realised by T uple κ ˆ M equipped with the coarse median induced by ś j µ j for κ large. Proof. By P roposition 2.2 , T uple ˚ ˆ M stabilises at some threshold κ 0 ě 0, and so T uple κ p ˆ M q ă T uple κ 1 p ˆ M q ă T uple κ 0 p ˆ M q ă T uple κ p ˆ M q for all κ 1 ě κ ě κ 0 . W e claim that T uple κ p ˆ M q is an approximate median subalgebra of ś j M j for κ ě κ 0 . Let x , y , z P T uple κ p ˆ M q . Then M φ µ i p x i , y i , z i q « C µ j p M φ x i , M φ y i , M φ z i q « ρ κ µ j p x j , y j , z j q for all arrows φ : i Ñ j . Therefor e p ś j µ j qp x , y , z q P T uple C ` ρ κ p ˆ M q ă T uple κ p ˆ M q , yielding the claim. N ote that any M –compatible set U satisfies U ă T uple κ p ˆ M q for some κ ě κ 0 . It follows that T uple κ p ˆ M q satisfies the desired ă –greatest condition, and hence yields a universal u.c. coarse median cone o ver M by P roposition 3.2 . 4 Quasigeodesic coarse median cones W e no w turn our attention to quasigeodesic u.c. coarse median cones, that is, where the apex is quasigeodesic. Call λ : p W , µ q M a universal quasigeodesic u.c. coarse median cone if every quasigeodesic u.c. coarse median cone ζ : p Z , ν q M factors through λ via a unique (up to closeness) CMP controlled map. W e shall consider the i nteraction between coarse medians and the Rips filtration. Given a coarse median µ on X , let ψ σ : p Rips σ X q 3 Ñ Rips σ X be the ternary operator co- inciding with µ on underlying sets. Lemma 4.1. A ssume that µ has upper control ρ . Then the function p Rips σ X q 3 Ñ Rips ρ σ X coinciding with µ on underlying sets is 1–Lipschitz for all σ ě 0 . Proof. Observe that p Rips σ X q 3 is isometric to Rips σ X 3 via the underlying identity . Sup- pose p x , y , z q , p x 1 , y 1 , z 1 q P X 3 are adjacent in Rips σ X 3 . Then d X 3 pp x , y , z q , p x 1 , y 1 , z 1 qq ď σ , hence d X p µ p x , y , z q , µ p x 1 , y 1 , z 1 qq ď ρ σ . The result follo ws. Lemma 4.2. Let p X , µ q be coarsel y geodesic coarse median space. Then for σ ě 0 large, ψ σ is a coarse median on Rips σ X . M oreo ver , ξ σ : p Rips σ X , ψ σ q Ñ p X , µ q is CMP . Proof. Let C be a coarse median constant for p X , µ q and ρ be an upper control for µ . W e may assume, without loss of generality , that ρ satisfies ρ p t q ě t for all t ě 0. B y Lemma 2.3 , there exists σ 0 ě 0 such that the underlying identity Rips ρ σ Ñ Rips σ X is a coarse equivalence for all σ ě σ 0 . Therefore , by Lemma 4.1 , ψ σ factors via controlled maps p Rips σ X q 3 Ñ Rips ρ σ X Ñ Rips σ X for σ ě σ 0 , and is hence controlled. B y choosing σ ě C , Coarse medians and univ ersal quasigeodesic cones 9 we obtain a coarse median space p Rips σ X , ψ σ q with coarse median constant 1. Note that µξ ˆ 3 σ “ ξ σ ψ σ b y construction, and so ξ σ is CMP . The follo wing result is a partial analogue of P roposition 3.2 Proposition 4.3. Let M : J ù CMed be a u.c. coarse median diagram. S uppose that ther e exists a ă –greatest element U among the coarsely geodesic M –compatible appr oximate me- dian subalgebr as of ś j M j . Then Rips σ U realises a univ ersal quasigeodesic u.c. coarse me- dian cone for σ large. Proof. Assume U Ď ś j M j satisfies the given ă –greatest property . Equip U with the coarse median µ 1 induced by ś j µ j . Since U is coarsely geodesic, by Lemma 2.3 , the map ξ σ : Rips σ U Ñ U coinciding with the underlying identity is a coarse equivalence for σ large. W e claim that π | U ˝ ξ σ : Rips σ U M is a universal quasigeodesic u.c. coarse median cone . Let ζ : p Z , ν q M be a q uasigeodesic u.c. coarse median cone. U sing the same argument as in the proof of Pr oposition 3.2 , we deduce that ζ factors through π | U via a unique (up to closeness) CMP controlled map h : p Z , ν q Ñ p U , µ 1 q . Z Rips σ U U M j Ð Ñ h Ð Ñ h 1 Ð Ñ ζ j Ð Ñ ξ σ Ð Ñ p π | U q j ˝ ξ σ Ð Ñ p π | U q j Since Z is quasigeodesic, Proposition 2.4 asserts the existence of a coarsely Lipschitz map h 1 : Z Ñ Rips σ U such that h « ξ σ h 1 . B y Lemma 4.2 , the coarse equivalence ξ σ is CMP , and so h 1 is CMP . Ther efore , ζ « π | U ˝ h « p π | U ˝ ξ σ q h 1 yields a desired factorisation. T o verify uniqueness, suppose that ζ « p π | U ˝ ξ σ q h 2 for some controlled map h 2 : Z Ñ Rips σ U . Then π | U ˝ p ξ σ h 2 q « π | U ˝ p ξ σ h 1 q and so, by arguing as in the proof of P roposition 3.2 , we deduce that ξ σ h 2 « ξ σ h 1 . Since ξ σ is a coarse equivalence, it follo ws that h 2 « h 1 . Thus, π | U ˝ ξ σ : Rips σ U M is a universal quasigeodesic u.c. coarse median cone. W e finally pro ve Theorem 1.1 . Theorem 4.4. Let M : J ù CMed be a u.c. coarse median diagram. Assume that its un- derlying u.c. diagram ˆ M admits a universal quasigeodesic u.c. cone λ : W ˆ M . Then W admits a canonical (up to closeness) coarse median ν such that λ : p W , ν q M is a univer - sal quasigeodesic u.c. coarse median cone. Proof. By Theorem 2.5 , lim ø J ˆ M exists and i s coarsely geodesic. Mor eo ver , T uple ˚ ˆ M sta- bilises to a coarsely geodesic space. By choosing κ ě 0 sufficiently large and appeal- ing to Theorem 3.3 , we deduce that T uple κ ˆ M is ă –greatest among coarsely geodesic M – compatible appro ximate median subalgebras of ś j M j . Ther efore , by Proposition 4.3 , we obtain a universal quasigeodesic u.c. coarse median cone ζ : Rips σ T uple κ ˆ M M for σ large , where the legs are restrictions of factor wise projections on underlying sets. Note that ζ is also a u.c. limit cone (since ξ σ is a coarse equivalence), and so there exists a unique (up to closeness) coarse equivalence f : W Ñ Rips σ T uple κ ˆ M such that λ « ζ f . In fact, f 10 Robert T ang is a quasi-isometr y since its domain and codomain are quasigeodesic. By Lemma 2.7 , we obtain a unique (up to closeness) coarse median ν on W for which f is CMP . 5 Bounded coarse inter val image In this section, we apply Theorem 4.4 to reco ver a theor em of B o wditch which states that any HHS admits a coarse median structure compatible with the coarse medians on the hyperbolic factor spaces [ Bo w18 ]; in particular , this yields a coarse median for the map- ping class group. W e shall not require the definition of HHSs here; instead, the inter ested reader may refer to [ BHS17a , BHS19 , Sis19 ] for further background. Our argument ap- plies mor e gener ally to families of uniformly coarse median spaces satisfying some pair- wise constraints modelled on the HHS–consistency axioms . W e do not assume any quasi- geodesicity (or coarse geodesicity) of the involved coarse median spaces . Consider a family tp C U , µ U qu U P S of unifor mly coarse median spaces, with index set S . H ere , we ass ume that all p C U , µ U q admit a comm on coarse median constant C and upper control ρ . The set S is equ ipped with a symmetr ic non-reflexive binar y relation K on S called orthogonality which we shall use to impose pairwise constraints on this family of spaces. For C U and C V where U K V , we impose no constraint; whereas for U M V (and distinct), we impose the follo wing: Bounded coarse inter val image. There is a constant B ě C ` ρ C such that for all U M V distinct, (up to swapping U and V ) there exists a function θ V U : C V Ñ C U and a point O U V P C V such that for all x V , y V P C V , we have that O U V P r x V , y V s B or θ V U x V « B θ V U y V . This condition is modell ed on the bounded geodesic image property in the setting where θ V U : C V Ñ C U is the subsurface projection between cur ve graphs [ MM00 ]. In that case , C V is Gromov hyperbolic and so the condition amounts to saying that if O U V does not lie near any geodesic from x V to y V in C V then the images of x V and y V under θ V U are uniformly close in C U . N o w , we define pair wise constraints on the family tp C U , µ U qu U P S . Fix K ě B . Whenever U K V , declar e R U V : “ C U ˆ C V . For U M V , declar e R U V : “ ␣ p x U , x V q P C U ˆ C V : x U « K θ V U x V or x V « K O U V ( . In either case, we equip R U V with the induced ℓ 8 –metric. These constraints simultane- ously capture the consisten cy conditions arising from the bounded geodesic image prop- erty and the Behrstock inequality (by setting θ V U constant) from the HHS Axioms . Let us verify that these R U V Ď C U ˆ C V are uniform appro ximate median subalgebras . Lemma 5.1. Ther e exists a constant R ě 0 depending on C , B , K , and ρ such for all distinct U , V P S , we have that p µ U ˆ µ V qp R 3 U V q Ď N R p R U V q . Proof. The U K V case is trivial, so we may assume U M V . Choose p x U , x V q , p y U , y V q , p z U , z V q P R U V and set m U “ µ U p x U , y U , z U q , m V “ µ V p x V , y V , z V q . B y L emma 2.8 , we de- duce m V P r x V , y V s C ` ρ C Ď r x V , y V s B , hence r x V , m V s B Ď r x V , y V s L for some L “ L p C , B , ρ q . If O U V P r x V , y V s L X r y V , z V s L X r z V , x V s L then by Lemma 2.10 , we deduce O U V « R m V for some R ě 0 depending on L , C , and ρ , hence p m U , m V q « R p m U , O U V q P R U V . W e may thus assume that O U V R r x V , y V s L Ě r x V , m V s B (up to cyclically permuting x V , y V , z V ). By Coarse medians and univ ersal quasigeodesic cones 11 the bounded coarse interval image assumption, we deduce θ V U m V « B θ V U x V « B θ V U y V . Since p x U , x V q , p y U , y V q belong to R U V , it follows that x U « K θ V U x V « B θ V U y V « K y U , and so m U « ρ p 2 K ` B q x U . Consequently , m U « R θ V U m V for R ě ρ p 2 K ` B q ` K ` B . Therefor e p m U , m V q « R p θ V U m V , m V q P R U V . Consequently , by Lemma 5.1 , each R U V admits a coarse median induced by µ U ˆ µ V ; moreo ver , the coarse median constants and upper contr ols can be chosen uniformly . F ol- lo wing [ T an , Section 6.1], we may define a u.c. coarse median diagram M : J ù CMed from the family tp C U , µ U qu U P S equipped with the pairwise constraints R U V as follo ws. Each vertex of J is labelled by either an element U P S or b y a distinct (unorder ed) pair U V ; the directed arro ws are given by U Ð U V Ñ V for each U V . The objects of M are the p C U , µ U q for each U P S , together with the p R U V , µ U ˆ µ V q for each (unorder ed) pair U , V P S distinct; the bonding maps are given by the (1–Lipschitz) factorwise projections C U Ð R U V Ñ C V . N ote that these bonding maps are uniformly CMP . Let us now apply this setup to HHSs. Her e, we are given a quasigeodesic space X , a fam- ily t C U u U P S of δ –hyperbolic spaces, for some δ ě 0, together with a uniformly coarsely Lipschitz family of projections t λ U : X Ñ C U u U P S . The index set S is equipped with an orthogonality r elation K . The HHS Axioms assert that pairwise projections λ U ˆ λ V factor through some pairwise constraint R U V Ď C U ˆ C V up to uniform error: for U K V there is no constraint, for U M V they satisfy either the Behrstock Inequality or the Bounded Geodesic Image property . Since the spaces C U are uniformly hyperbolic, we may endow them with uniform coarse medians µ U ; explicitly , let µ U p x U , y U , z U q be any δ –centr e of a geodesic triangle with corners x U , y U , z U P C U . Each R U V either equals C U ˆ C V or satisfies the Bounded Coarse Interval condition with uniform error and so , b y Lemma 5.1 , they admit uniform coarse medians induced by µ U ˆ µ V . W e thus obtain a u.c. coarse median diagram M : J ù CMed of the form given above . Corollary 5.2. Let X be an HHS and M : J ù CMed be the corresponding u.c. coarse median diagr am as given above. Then there exists a coarse median ν on X , unique up to closeness, such that λ : p X , ν q M is a universal quasigeodesic u.c. coarse median cone. Proof. By [ T an , Theorem 6.7], there is a universal quasigeodesic u.c. cone λ : X ˆ M whose legs are giv en by the λ U and λ U ˆ λ V . The result follo ws using Theorem 4.4 . R eferences [BHS17a] J. A. Behrstock, M. F . Hagen and A. Sisto , Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups , Geom. T opol. 21 (2017), no . 3, 1731–1804; MR3650081 [BHS19] J. A. Behrstock, M. F . Hagen and A. Sisto , Hierarchically hyperbolic spaces II: Combination theorems and the distance formula , Pacific J. Math. 299 (2019), no . 2, 257–338; MR3956144 [BM11] J. A. Behrstock and Y . N. Minsky , Centroids and the rapid decay property in mapping class groups, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 765–784; MR2855801 12 Robert T ang [Bo w13a] B . H. Bo wditch, Coarse median spaces and groups , P acific J. 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