Coarse separation and splittings in hyperbolic groups

Coarse separation and splittings in h yp erb olic groups Oussama Bensaid, An thony Genev ois, and Romain T essera Marc h 19, 2026 Abstract W e study coarse separation in one-ended hyperb olic groups from a quantitativ e p oint of view, fo cusing on the v olume growth of separating subsets. W e prov e that a one-ended h yp erb olic group that is not virtually a surface group is coarsely separable by a subset of sub exponential gro wth if and only if it splits ov er a virtually cyclic subgroup. T o do so, w e sho w that sufficiently large thick ened spheres are hard to cut, in the sense that their cut- sets hav e exp onen tial size, a result of independent in terest. As an application, we obtain a polynomial low er b ound on the separation profile of one-ended h yp erbolic groups that do not split ov er a t wo-ended subgroup. W e also apply our criterion to graph pro ducts of finite groups, giving a com binatorial characterisation of when such graph products are coarsely separable by a subset of subexp onen tial gro wth. Con ten ts 1 In tro duction 1 2 Preliminaries 5 2.1 Coarse separation and cuts . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Hyp erbolic groups and their b oundaries . . . . . . . . . . . . . . . . . . 7 2.2.1 Gromo v b oundary . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Uniform distortion of spheres in one-ended hyperb olic groups . . 10 2.2.3 Shado ws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Pro of of the main theorem 14 4 Applications to graph pro ducts 16 4.1 Graph pro ducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Hyp erbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Pro of of Theorem 1.10 and examples . . . . . . . . . . . . . . . . . . . . 22 References 23 1 In tro duction Separation phenomena in groups ha v e long play ed a cen tral role in geometric group theory , as they often reflect the presence of algebraic splittings. The starting p oin t is Stallings’ theorem [Sta71]: a finitely generated group has more than one end if and only if it splits o ver a finite subgroup. Equiv alen tly , a Ca yley graph of the group can b e separated into at least tw o deep comp onen ts by a finite subset if and only if the group 2010 Mathematics subje ct classific ation. Primary 20F65. Secondary 20F69. K ey wor ds and phrases. Hyperb olic groups, graph pro ducts, splitting, coarse separation. 1 splits o ver a finite subgroup. In this sense, a coarse separation prop ert y already detects a non trivial splitting, and hence an algebraic feature that is in v arian t under quasi-isometry . This philosoph y has since b een dev elop ed in several directions. A first generalisation of this principle was obtained b y Dun woo dy–Sw enson [DS00], who extended the subgroup- theoretic side of Stallings’ theorem from finite subgroups to finitely generated virtually p olycyclic co dimension-one subgroups. In particular, if G is one-ended and not virtually a surface group, then G splits ov er a t wo-ended subgroup if and only if it con tains an infinite cyclic co dimension-one subgroup. Recall that a subgroup H ≤ G is c o dimension- one if some finite neighbourho o d of a copy of H in a Ca yley graph of G separates that Ca yley graph in to at least t w o deep components. Another generalisation w as later pro ved by P apasoglu [P ap05], who extended the geometric side of Stallings’ theorem b y replacing finite separating sets with quasi- lines: he sho wed that a one-ended finitely presen ted group that is not commensurable to a surface group splits ov er a tw o-ended subgroup if and only if its Ca yley graph is separated by a quasi-line. In particular, admitting such a splitting is a quasi-isometry in v arian t. In the same spirit, and follo wing [BGT24], w e study coarse separation b y families rather than individual subsets. W e do so from a quantitativ e p oin t of view, focusing on the v olume growth of the separators rather than on their quasi-isometry type. W e pro ve Theorem 1.1. L et G b e a hyp erb olic gr oup. A ssume that G is one-ende d and not virtual ly a surfac e gr oup. Then G is c o arsely sep ar able by a family of sub exp onential gr owth if and only if it splits over a virtual ly cyclic sub gr oup. W e refer to Section 2.1 for the definitions of coarse separation and of the volume gro wth of a family of subsets. The exclusion of virtually surface groups is necessary . Indeed, virtually surface groups are quasi-isometric to H 2 , so they are coarsely separable b y subsets of linear growth, for instance by thick ened geo desics. Ho wev er, there exist h yp erb olic triangle groups that are virtually surface groups but do not split o ver t wo- ended subgroups [Bo w98]. W e also note that, in the hyperb olic setting, the previous discussion has a b oundary coun terpart: Bowditc h [Bow98] show ed that splittings o ver t wo-ended subgroups are detected by lo cal cut p oints in the boundary . This p oin t of view will b e essen tial in the pro of. Coarse separation and splittings. Theorem 1.1 exhibits a form of rigidity , pro v- ing that, for one-ended h yp erbolic groups that are not virtually surface groups, coarse separation by a family of sub exp onen tial growth can o ccur only when there is a strong form of algebraic separation, namely a splitting ov er a virtually cyclic subgroup. In a forthcoming article [BGT26], we will highlight a similar phenomenon for righ t-angled Artin groups b y proving that a right-angled Artin group is coarsely separable by a family of sub exp onen tial gro wth if and only if it splits ov er an ab elian subgroup. Then, it is natural to wonder to which exten t this observ ation is common among finitely generated groups. Question 1.2. Under which reasonable as sumptions do es a finitely generated group coarsely separable by a family of sub exp onen tial growth (virtually) split ov er a subgroup of sub exponential gro wth? In view of Theorem 1.1 and [BGT26], it is reasonable to exp ect p ositive answers to the follo wing questions: Question 1.3. If a relativ ely h yp erb olic group is coarsely separable by a family of sub exponential gro wth, do es it virtually split ov er a virtually cyclic subgroup? Question 1.4. If a righ t-angled Coxeter group, or more generally a virtually co com- pact sp ecial group, is coarsely separable b y a family of sub exp onen tial growth, do es it virtually split ov er a virtually ab elian subgroup? 2 Question 1.2 can b e decomposed into tw o sub questions. First, one can ask whether the geometric separation can occur through some subgroup. A priori, there is no reason for a coarsely separating family to b e close in any sense to a subgroup. In fact, one can alw ays enrich a coarsely separating family in order to mak e it as different as w e w ant from a subgroup. Nevertheless, we can ask: if a giv en finitely generated G is coarsely separable b y a family of sub exp onen tial growth, can w e find a co dimension-one subgroup H ≤ G of sub exponential gro wth? Note that one may also define a codimension-on e subgroup H ≤ G as a subgroup such that the Schreier graphs Sc h( G, H ) , constructed from finite generating sets of G , are m ulti-ended. One also sa ys that G semi-splits over H . A degenerate case of our question deals with finitely generated groups that ha ve sub ex- p onen tial gro wth themselves. Can w e alwa ys find codimension-one subgroups in suc h groups? Recall from [Sag95, NR98] that a finitely generated group con tains a co dimension- one subgroup if and only if it admits an action on a median graph (or equiv alently , a CA T(0) cube complex) with un b ounded orbits. When such an action do es not exist, one sa ys that the group satisfies Pr op erty (FW) . Question 1.5. Do es there exist a group of intermediate gro wth satisfying (FW)? A ccording to [NR98], groups satisfying Kazhdan’s Property (T) pro vide a source of examples of groups satisfying (FW), and in fact the almost exclusiv e source of s uc h examples. But they are of no help here, since groups of sub exponential growth are amenable and that the only amenable groups satisfying (T) are the finite groups. Our second subquestion deals with a problem that had attracted a lot of atten tion a couple of dedaces ago: assuming that a finitely generated group G semi-splits o ver a subgroup H , can w e find a (virtual) splitting of G ov er a subgroup related to H ? W e refer the reader to the surv ey [W al03] for more information on this general problem. Here, w e are interested in the case where H has sub exponential growth. Notice that, even under this restriction, a p ositiv e answ er to the question is not reasonable. F or instance, it is kno wn that the Grigorc huk group admits co dimension-one subgroups, necessarily of sub exponential growth, but, since the Grigorch uk group is torsion, there is no (virtual) splitting what so ev er. On the other hand, it is w orth mentioning that, for p olynomial gro wth (i.e. virtually nilp otent groups), the problem is rather well-understoo d thanks to the algebraic torus theorem [DS00] (which, more generally , deals with semi-splittings o ver p olycyclic subgroups). Since the situation seems to b e more tame for p olynomial gro wth, it is then natural to ask: Question 1.6. Does a finitely generated group coarsely separable by a family of p oly- nomial gro wth virtually split o ver a subgroup of p olynomial gro wth? In fact, in view of Stalling’s theorem ab out the n umber of ends and of P apasoglu’s theorem [P ap05] ab out coarse separation by quasi-lines, it is ev en more natural to ask: Question 1.7. Do es a finitely generated group coarsely separable b y an n -dimensional quasiflat virtually split ov er a virtually ab elian subgroup of rank n ? Strategy . In fact, we show that if G is a one-ended h yp erb olic group that is not virtually a surface group and do es not split ov er a virtually cyclic subgroup, then there exists a constan t D ≥ 0 suc h that the D -neighbourho ods of all sufficien tly large spheres are “hard to cut”, in the sense that their cut sets ha ve uniformly exponential growth, see Section 2.1 for a definition of the cut of a subgraph. Our main result, from which Theorem 1.1 follows, is the follo wing theorem, which is of indep enden t in terest. 3 Theorem 1.8. L et G b e a one-ende d hyp erb olic gr oup that is not virtual ly a surfac e gr oup and do es not split over any two-ende d sub gr oup. Then ther e exists t 0 ≥ 1 such that, for any t ≥ t 0 and any ε ∈ (0 , 1) , ther e exist c onstants λ > 0 and C > 0 such that the fol lowing holds: for every g ∈ G and every sufficiently lar ge inte ger n , cut ε  S G ( g , n ) + t  ≥ C e λn . A rough outline of the pro of is as follo ws. First, note that, since G is one-ended, a thic k ened sphere S G ( g , n ) + t is a connected subgraph of G , for t large enough. Let Z n ⊂ S G ( g , n ) + t b e an ε -cut, and suppose tow ards contradiction that Z n is “small” . A first step sho ws that S G ( g , n ) + t \ Z n m ust contain t wo connected components with p oin ts x n , y n lying v ery far from Z n . W e then pass from the sphere to the b oundary b y means of shado ws: the shado ws sh ( x n ) and sh ( y n ) lie in differen t connected components of ∂ G \ sh ( Z n ) , so that sh ( Z n ) separates the b oundary . Moreo v er, they lie “far” from sh ( Z n ) . W e then use a result of Lazaro vic h–Margolis–Mj [LMM24], see Theorem 2.12, whic h pro vides, inside any minimal closed separator of the boundary , a uniform density prop ert y at all small scales. By iterating this density prop ert y , we obtain exponentially man y well-separated points in sh ( Z n ) . Because they are w ell separated, we can pull this separated family bac k to a separated family inside Z n , forcing Z n itself to b e “big”, a con tradiction. Theorem 1.1 follows from Theorem 1.8 by a result from [BGT24], see Theorem 2.7. The idea is as follo ws. First, thic k ened spheres form a p ersistent family , in the sense that t wo neighbouring thick ened spheres in tersect in a p ositiv e prop ortion. Therefore, if t wo thick ened spheres are separated b y some subset, then, b y moving from one sphere to the other through neigh b ouring spheres, one finds a thick ened sphere whose in tersection with the separating subset is a cut-set. Since suc h cut-sets ha ve exponential size b y Theorem 1.8, it follows that the separating subset itself must hav e exp onen tial growth. Applications of our tec hniques. As a consequence of Theorem 1.8, suc h h yp erb olic groups ha ve a separation profile that is b ounded b elo w b y a p olynomial: Theorem 1.9. L et G b e a one-ende d hyp erb olic gr oup that is not virtual ly a surfac e gr oup. I f G do es not split over any two-ende d sub gr oup, then ther e exist c onstants C > 0 and ε > 0 such that, for every n ∈ N , sep G ( n ) ≥ C n ε . W e refer to Section 2.1 for the definition of the separation profile of a group. W e note that this result was also obtained recen tly b y Hume–Macka y [HM25, Theorem 1.11] using differen t metho ds: instead of pro ving that spheres are hard to cut, they use techniques from [Mac19] to embed “sufficiently thic k” round trees. Another application of Theorem 1.1 concerns graph pro ducts of finite groups. Our main result in this direction is the following. Theorem 1.10. L et Γ b e a finite □ -fr e e gr aph and G a c ol le ction of finite gr oups indexe d by V (Γ) . The gr aph pr o duct Γ G is c o arsely sep ar able by a family of sub exp onential gr owth if and only if Γ c ontains a sep ar ating sub gr aph that c an b e written as a join A ∗ B , wher e A is a c omplete gr aph and B is either empty or c onsists of two non-adjac ent vertic es lab el le d by Z 2 . A brief outline of the pro of is as follows. First, we determine exactly when a graph pro duct is virtually cyclic or virtually a surface group. Then, we characterise splittings o ver virtually cyclic subgroups in terms of separating subgraphs of the defining graph. Finally , we combine this c haracterisation with Theorem 1.1 and with the classification of virtually cyclic graph pro ducts to obtain the theorem. 4 A c knowledgemen ts The first-named author ackno wledges supp ort from the FWO and F.R.S.-FNRS under the Excellence of Science (EOS) programme (pro ject ID 40007542). 2 Preliminaries If ( X , d ) is a metric space, w e denote by B ( x, r ) (resp. S ( x, r ) ) the closed ball (resp. the sphere) of radius r : B ( x, r ) := { z ∈ X | d ( x, z ) ≤ r } , S ( x, r ) := { z ∈ X | d ( x, z ) = r } . If A ⊆ X is a subset, w e denote by | A | ∈ N ∪ { + ∞} its cardinalit y , and b y A + r its r -neigh b ourhoo d: A + r := { x ∈ X | d ( x, A ) ≤ r } . 2.1 Coarse separation and cuts W e start b y recalling some definitions from [BGT24]. Since w e will only w ork with graphs in this pap er, w e restrict all definitions to graphs. W e begin with the central notion of coarse separation. Definition 2.1. Let X b e a graph and Z a collection of subgraphs. A connected subgraph Y ⊂ X is c o arsely sep ar ate d by Z if there exists L ≥ 0 such that for every D ≥ 0 , there is some Z ∈ Z such that Y \ Z + L has at least tw o connected comp onents with p oin ts at distance ≥ D from Z . W e will call L the thickening c onstant of Z . Definition 2.2. Let X b e a graph of b ounded degree, and let S b e a family of subgraphs of X . W e define its gro wth by V S ( r ) := sup s ∈ Y , Y ∈S | B X ( s, r ) ∩ Y | . W e say that S has exp onential volume gr owth if lim sup r →∞ 1 r log V S ( r ) > 0 . Otherwise, we sa y that S has sub exp onential volume gr owth . W e denote by M exp the class of b ounded degree graphs suc h that an y coarsely separating family of subgraphs m ust hav e exp onen tial volume gro wth. The following definitions of a “cut” of a subgraph and of the separation profile of a graph are due to Benjamini–Schramm–Timár [BST12] and quantify the connectivity of the graph. Definition 2. 3. Let X b e a connected graph of b ounded degree, and let δ ∈ (0 , 1) . Let A ⊂ X b e a finite subgraph, and let S ⊂ A . Then S is a δ -cut of A if every connected comp onen t of A \ S has size ≤ δ | A | . W e denote by cut δ ( A ) the minimal size of a δ -cut of A . The sep ar ation pr ofile of X is the function sep X ( n ) := sup { cut 1 2 ( S ) : S ⊂ X , | S | ≤ n } , n ∈ N . In the definition of the separation profile, the constan t 1 / 2 can b e replaced by any δ ∈ (0 , 1) without changing the function up to the following equiv alence: t w o functions f , g : N → [0 , ∞ ) are said to b e equiv alent if there exists a constan t C ≥ 1 suc h that 1 C g ( n ) ≤ f ( n ) ≤ C g ( n ) for all n ∈ N . 5 Lemma 2.4. L et X b e a c onne cte d gr aph of b ounde d de gr e e, and δ ∈ (0 , 1) . Set δ ′ = min( δ 4 , 1 − δ 4 ) . Then for any subset E ⊂ X and any δ -cut S ⊂ E of E , one of the fol lowing holds (1) | S | ≥ δ ′ | E | ; (2) E \ S c an b e p artitione d E \ S = A ⊔ B , wher e e ach of A and B is a union of c onne cte d c omp onents of E \ S , | A | ≥ δ ′ | E | and | B | ≥ δ ′ | E | . Pr o of. Supp ose | S | < δ ′ | E | . Let ( A i ) i =1 ,...,n b e the connected comp onen ts of E \ S . So n X i =1 | A i | > (1 − δ ′ ) | E | . Supp ose that the connected comp onen ts are ordered such that | A i | ≥ | A i +1 | . If | A 1 | ≥ δ ′ | E | , then n X i =2 | A i | ≥ (1 − δ ′ ) | E | − δ | E | ≥ δ ′ | E | , b ecause | A 1 | ≤ δ | E | . Therefore we hav e the partition. Otherwise, let p b e the smallest in teger such that P p i =1 | A i | ≥ δ ′ | E | . So P p i =1 | A i | ≤ 2 δ ′ | E | . n X i = p +1 | A i | ≥ (1 − δ ′ ) | E | − 2 δ ′ | E | ≥ δ ′ | E | , b ecause δ ′ ≤ 1 4 . Definition 2.5. Let X be a connected graph of b ounded degree, and let r 0 ≥ 0 and α > 0 . A family { A x ( r ) } x ∈ X, r ≥ r 0 of subgraphs of X is called α -p ersistent if, for ev ery r ≥ r 0 , the following hold: • for all x ∈ X , A x ( r ) ⊂ B X ( x, 4 r ) ; • for all x, y ∈ X , | A x ( r ) | = | A y ( r ) | ; • for ev ery pair of neighbouring v ertices x, y ∈ X , | A x ( r ) ∩ A y ( r ) | ≥ α | A ( r ) | . Example 2.6. Let X b e a connected vertex-transitiv e graph of v alency d , and let t ≥ 1 . F or every x ∈ X and every r ≥ t , set A x ( r ) := S ( x, r ) + t . Then the family { A x ( r ) } x ∈ X, r ≥ t is α -p ersistent, with α = 1 | B ( t ) | , see [BGT24, Example 2.14]. W e end this subsection with the follo wing result, whic h is a special case of [BGT24, Theorem 3.5] and motiv ates the definition of p ersisten t families. Theorem 2.7. L et X b e a c onne cte d gr aph of b ounde d de gr e e, and let { A x ( r ) } x,r b e an α -p ersistent family. If ther e exist c onstants C, β > 0 such that, for every x ∈ X and every sufficiently lar ge r , cut δ ( A x ( r )) ≥ C e β r , wher e δ = 1 − α 2 , then X ∈ M exp . 6 2.2 Hyp erb olic groups and their b oundaries 2.2.1 Gromo v b oundary Let G b e a non-elementary h yp erb olic group, i.e. an infinite hyperb olic group that is not virtually cyclic. W e also denote b y G its Cayley graph with resp ect to a fixed finite generating set, and let δ be its hyperb olicity constan t, in the sense that ev ery geo desic triangle in G is δ -slim. The Gromo v b oundary ∂ G admits a visual metric ρ : there exist a > 1 , C ≥ 1 such that, for all ξ , η ∈ ∂ G and for any bi-infinite geo desic γ connecting them, w e hav e 1 C a − d ( e,γ ) ≤ ρ ( ξ , η ) ≤ C a − d ( e,γ ) . Giv en x, y ∈ G and ξ ∈ ∂ G , w e denote b y [ x, y ] a geodesic segment from x to y , and b y [ x, ξ ) a geo desic ray from x to ξ . The follo wing prop osition is standard. W e include a pro of for completeness. Prop osition 2.8. Ther e exist c onstants C 1 , C 2 > 0 such that for any ξ , η ∈ ∂ G , any n ∈ N , and any x, y ∈ S n for which ther e exist ge o desic r ays [ e, ξ ) and [ e, η ) p assing thr ough x and y , r esp e ctively, we have ρ ( ξ , η ) ≤ C 2 a − n a d ( x,y ) 2 . (1) In p articular, for every ω > 0 , if ρ ( ξ , η ) ≥ ω then d ( x, y ) ≥ 2 n + 2 log a  ω C 2  . (2) Mor e over, if d ( x, y ) ≥ d 0 := 20 δ + 1 , then C 1 a − n a d ( x,y ) 2 ≤ ρ ( ξ , η ) . (3) In p articular, if d ( x, y ) ≥ d 0 , then ρ ( ξ x , ξ y ) > 0 for any ξ x ∈ ∂ G (r esp. ξ y ) extending [ e, x ] (r esp. [ e, y ] ). Pr o of. F or the first inequalit y , we refer to [BGT24, Claim 3.19 (i)]. Supp ose that d ( x, y ) ≥ 20 δ +1 , and let ˜ γ be a bi-infinite geo desic with endp oin ts ξ , η . The (generalised) triangle [ e, ξ ] ∪ ˜ γ ∪ [ e, η ] is 5 δ -slim, see [CDP90, Prop osition 2.2], so x ∈ ( ˜ γ ∪ [ e, η ]) +5 δ . If x ∈ [ e, η ] +5 δ , w e w ould get d ( x, y ) ≤ 10 δ . Therefore x ∈ ˜ γ +5 δ , and similarly y ∈ ˜ γ +5 δ . Let x ′ , y ′ ∈ ˜ γ suc h that d ( x, x ′ ) ≤ 5 δ , d ( y , y ′ ) ≤ 5 δ . Let γ b e a geodesic segment from x to y , and let m b e its midpoint. Let α be a geo desic segmen t from x to y ′ . In the geo desic triangle γ ∪ α ∪ [ y, y ′ ] , since d ( m, y ) ≥ 10 δ , then d ( m, [ y , y ′ ]) ≥ 5 δ . Therefore, there exists m 1 ∈ α suc h that d ( m, m 1 ) ≤ δ . Now consider the geo desic triangle α ∪ [ x ′ , y ′ ] ∪ [ x, x ′ ] . W e hav e d ( m 1 , x ) ≥ d ( m, x ) − δ ≥ 9 δ , so d ( m 1 , [ x, x ′ ]) ≥ 4 δ . Hence there exists m ′ ∈ [ x ′ , y ′ ] suc h that d ( m 1 , m ′ ) ≤ δ . Therefore d ( m, m ′ ) ≤ 2 δ . No w consider the geo desic triangle [ e, x ] ∪ γ ∪ [ e, y ] . Again by δ -slimness, m ∈ ([ e, x ] ∪ [ e, y ]) + δ . W e may assume that there exists z ∈ [ e, x ] such that d ( m, z ) ≤ δ . Since d ( x, y ) 2 = d ( x, m ) ≤ d ( x, z ) + d ( z , m ) ≤ d ( x, z ) + δ, w e get that d ( x, z ) ≥ d ( x,y ) 2 − δ . Therefore d ( e, z ) ≤ n − d ( x,y ) 2 − δ , and d ( e, m ) ≤ n − d ( x, y ) 2 + 2 δ. 7 Finally , we get d ( e, ˜ γ ) ≤ d ( e, m ′ ) ≤ d ( e, m ) + d ( m, m ′ ) ≤ n − d ( x, y ) 2 + 4 δ. Therefore, ρ ( ξ , η ) ≥ 1 C a − d ( e, ˜ γ ) ≥ 1 C a − ( n − d ( x,y ) 2 +4 δ ) = a − 4 δ C a − n a d ( x,y ) 2 . Set C 1 := a − 4 δ C . By [Co o93, Thm 7.2], there exist α > 0 and K ≥ 1 such that for any g ∈ G and any in teger n , 1 K e αn ≤ | B ( g , n ) | ≤ K e αn . (4) Since G is non-elemen tary , therefore non-amenable, there exists K ′ ≥ 1 suc h that for an y g ∈ G and an y integer n , 1 K ′ e αn ≤ | S ( g , n ) | ≤ K ′ e αn . (5) T o simplify notation, w e set B n := B ( e, n ) and S n := S ( e, n ) . As a consequence of Lemma 2.4, the next proposition sho ws that, unless an ε -cut of a thick ened sphere is exp onen tially large, it must lea ve tw o connected comp onen ts con taining p oin ts very far from the cut. Prop osition 2.9. L et G b e a non-elementary hyp erb olic gr oup, and let α, K, K ′ b e the c onstants given by (4) and (5) . L et ε ∈ (0 , 1) and t ≥ 1 . Then ther e exists Θ > 0 such that, for every inte ger n and every ε -cut Z n ⊂ S + t n , one of the fol lowing holds: (1) ther e exist x n , y n in differ ent c onne cte d c omp onents of S + t n \ Z n such that d ( { x n , y n } , Z n ) > n 2 ; (2) | Z n | ≥ Θ e α 2 n . Pr o of. Let n b e an in teger, and let S + t n \ Z n = A ⊔ B b e a partition of the connected comp onen ts of S + t n \ Z n as in Lemma 2.4. Denote the connected comp onen ts by A = ⊔ i ∈ I A i and B = ⊔ j ∈ J B j . If A i ⊂ Z + n 2 n for an y i ∈ I , then A ⊂ Z + n 2 n , and ε ′ | S + t n | ≤ | A | ≤ | Z n | × sup g ∈ G | B ( g , n/ 2) | ≤ | Z n | K e α n 2 , where ε ′ = min( ε 4 , 1 − ε 4 ) . Since 1 K ′ e αn ≤ | S n | ≤ | S + t n | , w e get | Z n | ≥ Θ e α 2 n , for Θ = ε ′ K K ′ . Otherwise, let i 0 ∈ I suc h that A i 0 ⊂ Z + n 2 n . If (1) is not satisfied, then for an y j ∈ J , B j ⊂ Z + n 2 n . Therefore B ⊂ Z + n 2 n and the same argument implies once again that | Z n | ≥ Θ e α 2 n . 8 F or every n ∈ N , let π n : G \ B n − 1 → S n b e the pro jection that asso ciates to g the in tersection of S n with [ e, g ] , with resp ect to a fixed choice of geo desic segments [ e, g ] from e . Prop osition 2.10. L et G b e a non-elementary hyp erb olic gr oup. (i) F or d 0 > 0 as in Pr op osition 2.8, if γ 1 , γ 2 ar e asymptotic ge o desic r ays emanating fr om e , then for any t ≥ 0 , d ( γ 1 ( t ) , γ 2 ( t )) ≤ d 0 . (ii) Ther e exists a c onstant D ≥ 0 such that every g ∈ G is at distanc e at most D fr om a ge o desic r ay emanating fr om e . (iii) F or every x, y ∈ G \ B n − 1 , d ( π n ( x ) , π n ( y )) ≤ d ( x, y ) + 2 δ. Pr o of. P art ( i ) follows directly from (3), and see [BM91, Lemma 3.1] for ( ii ) . ( iii ) Let x, y ∈ G \ B n − 1 , and let α be the geodesic segmen t [ x, y ] . Consider the geo desic triangle [ e, x ] ∪ α ∪ [ y , e ] . If π n ( x ) ∈ [ e, y ] + δ or π n ( y ) ∈ [ e, x ] + δ , then it is easy to c heck that d ( π n ( x ) , π n ( y )) ≤ 2 δ. Otherwise, let x ′ , y ′ ∈ α b e such that d ( π n ( x ) , x ′ ) ≤ δ and d ( π n ( y ) , y ′ ) ≤ δ . Since d ( x ′ , y ′ ) ≤ d ( x, y ) , we conclude b y the triangle inequality . Prop osition 2.11. L et G b e a non-elementary hyp erb olic gr oup, d 0 > 0 b e as in Pr op o- sition 2.8, n ∈ N , and x, y ∈ S n . Then the fol lowing holds. (i) If x lies in some ge o desic r ay [ e, ξ ) and y lies in some ge o desic r ay [ e, η ) , then any c ontinuous p ath ˜ γ ⊂ ∂ G fr om ξ to η c an b e pul le d-b ack to a d 0 -p ath in S n : ther e exists a se quenc e of p oints x = x 0 , x 1 , . . . , x k = y in S n such that, for any i , d ( x i , x i +1 ) ≤ d 0 , and x i lies in some ge o desic r ay [ e, ξ i ) , with ξ i ∈ ˜ γ . (ii) If G is one-ende d, set t 0 := 3 max { D , d 0 } , wher e D is the c onstant fr om Pr op osition 2.10 and d 0 is the c onstant fr om Pr op o- sition 2.8. Then for every t ≥ t 0 and n ∈ N , S + t n is c onne cte d. (iii) L et t ≥ t 0 . If ther e exists a p ath of length d fr om x to y in G \ B n − 1 , then ther e exists a p ath fr om x to y in S + t n of length ≤ (1 + 2 δ ) d . Pr o of. ( i ) Let ˜ γ : [0 , L ] → ∂ G b e a contin uous path. By compactness of ˜ γ ([0 , L ]) , there exists a partition 0 = t 0 ≤ t 1 ≤ · · · ≤ t k = L suc h that for an y i , ρ  ˜ γ ( t i ) , ˜ γ ( t i +1 )  < 1 C 1 a − n , where C 1 is the constant from Prop osition 2.8. F or each i , let x i b e the intersection of a geo desic ra y from e to ˜ γ ( t i ) with S n . By (3), w e obtain d ( x i , x i +1 ) ≤ d 0 . ( ii ) Let t ≥ t 0 and let n ∈ N . Let x, y ∈ S + t n . Cho ose x ′ , y ′ ∈ S n suc h that d ( x, x ′ ) ≤ t and d ( y , y ′ ) ≤ t. 9 There exist geo desic ra ys c, c ′ starting from e , and p oin ts x ′′ ∈ c , y ′′ ∈ c ′ suc h that d ( x ′ , x ′′ ) ≤ D and d ( y ′ , y ′′ ) ≤ D. Up to replacing D by 2 D in the previous inequalities, w e may assume that x ′′ , y ′′ ∈ S n . Since t ≥ t 0 , w e hav e d 0 ≤ t and 2 D ≤ t. By considering a path in ∂ G from c (+ ∞ ) to c ′ (+ ∞ ) and applying ( i ) to this path, we obtain a d 0 -path in S n from x ′′ to y ′′ . Replacing eac h step of this d 0 -path by a geo desic segmen t in G , and since d 0 ≤ t , w e obtain a path con tained in S + t n . Finally , by adding the geo desic segmen ts [ x, x ′ ] , [ x ′ , x ′′ ] , [ y ′′ , y ′ ] , [ y ′ , y ] , and since 2 D ≤ t , these segments also lie in S + t n . Hence we obtain a path in S + t n from x to y . ( iii ) Let γ b e a path of length d from x to y in G \ B n − 1 . By Proposition 2.10, π n ( γ ) is a (1 + 2 δ ) -path in S n from x to y . By replacing each step of this (1 + 2 δ ) -path by a geo desic segmen t in G , and since 1 + 2 δ ≤ t 0 ≤ t , w e obtain a path con tained in S + t n . Moreo ver, its length is ≤ (1 + 2 δ ) d . W e end this subsection by citing a result of Lazaro vich–Margolis–Mj [LMM24, Claims 4.2 and 4.3], which describ es minimal closed separators in the b oundary and sho ws that they satisfy a uniform densit y prop ert y . This result will be crucial in the proof of Theorem 1.8. Theorem 2.12. L et G b e a one-ende d hyp erb olic gr oup that do es not split over a two- ende d sub gr oup, and let ρ b e a visual metric on ∂ G . Then ther e exists a c onstant λ ∈ (0 , 1) such that the fol lowing holds. If Z ⊆ ∂ G sep ar ates ∂ G , and ξ 1 , ξ 2 ∈ ∂ G \ Z b elong to differ ent c onne cte d c omp onents of ∂ G \ Z , then ther e exists a minimal close d subset (with r esp e ct to inclusion) Z ′ of Z that sep ar ates ξ 1 and ξ 2 . Mor e over, for any ξ ∈ Z ′ and any 0 < r < ρ  { ξ 1 , ξ 2 } , Z ′  , Z ′ ∩  ¯ B ρ ( ξ , r ) \ B ρ ( ξ , λr )   = ∅ , wher e ¯ B ρ and B ρ denote the close d and op en b al ls, r esp e ctively, in ( ∂ G, ρ ) . 2.2.2 Uniform distortion of spheres in one-ended h yp erb olic groups By Prop osition 2.11, for ev ery t ≥ t 0 and ev ery n ∈ N , S + t n is connected. F or x, y ∈ S + t n , w e define d S + t n ( x, y ) to be the length of a shortest path in S + t n joining x to y . The aim of this subsection is to sho w that the family ( S + t n , d S + t n ) n ∈ N is uniformly exp onen tially distorted, indep enden tly of n : Prop osition 2.13. L et G b e a one-ende d hyp erb olic gr oup, and let t 0 b e as in Pr op osi- tion 2.11. Then for every t ≥ t 0 , ther e exist c onstants µ 1 , µ 2 , A 1 , A 2 > 0 such that the fol lowing holds. F or every n > t , and every x, y ∈ S + t n such that d ( x, y ) ≥ 9 δ + 6 t , A 1 e µ 1 d ( x,y ) − 2 t ≤ d S + t n ( x, y ) ≤ A 2 e µ 2 d ( x,y ) . The pro of will follo w from the next lemmas. The first one is standard. Lemma 2.14. L et X b e a δ -hyp erb olic gr aph, wher e δ > 0 , and let t ≥ 0 . Then for every R > t , and every o, x, y ∈ X such that x, y ∈ S ( o, R ) , the fol lowing holds. If d ( x, y ) ≥ 9 δ + 4 t , then any p ath fr om x to y in the c omplement of B ( o, R − t ) has length ≥ 2 d ( x,y ) 4 δ − 1 . 10 Pr o of. Set d := d ( x, y ) , let α b e a geo desic segmen t [ x, y ] and let m b e its midpoint. By considering the geo desic triangle [ o, x ] ∪ α ∪ [ y, o ] and using its δ -slimness, we sho w, as in the pro of of Prop osition 2.8, that d ( o, m ) ≤ R − d 2 + 2 δ. By the triangle inequality , and since d > 8 δ + 4 t , we obtain B  m, d 4  ⊆ B ( o, R − t ) . Let γ be an y path from x to y contained in X \ B ( o, R − t ) . Then γ av oids B ( m, d 4 ) , and the conclusion follows from [BH13, Chap. 3, Prop. 1.6]. F or the other inequalit y , w e need one-endedness. Before that, let us recall the follow- ing. Let D b e the constan t of Prop osition 2.10. In [BM91], Bestvina–Mess introduced the following property ( ‡ M ) , for M > 0 . W e say G satisfies the prop ert y ( ‡ M ) if there exists a constant L > 0 , suc h that for ev ery R > 0 and for ev ery pair of p oin ts x, y ∈ S R with d ( x, y ) ≤ M , there exists a path in G \ B R − D from x to y of length at most L . Bestvina–Mess sho w that if ( ‡ M ) fails to hold for some M > 0 , then the b oundary m ust con tain a cut point. How ever, it w as sho wn later b y Bo wditc h, Levitt, and Sw arup in [Bo w99, Lev98, Swa96] that the b oundary of a one-ended h yp erb olic group can never con tain a cut p oint. Therefore, ( ‡ M ) holds for ev ery M > 0 for one-ended hyperb olic groups. As a consequence, we ha ve Lemma 2.15. ([HR25, L emma 4.1]) L et G b e a one-ende d hyp erb olic gr oup. Ther e exists λ > 0 such that the fol lowing holds. If c and c ′ ar e ge o desic r ays emanating fr om e , and α is a p ath of length at most  fr om c to c ′ in the c omplement of the b al l B r , then ther e exists a p ath β of length at most λ fr om c to c ′ in the c omplement of the b al l B r +1 . One can therefore “push” outw ard a path b et ween p oin ts in S n , which will yield the upp er bound. Lemma 2.16. L et G b e a one-ende d hyp erb olic gr oup, and let t 0 b e as in Pr op osi- tion 2.11. Ther e exist c onstants µ, C > 0 such that the fol lowing holds. F or every n ∈ N , for every t ≥ t 0 , and every x, y ∈ S n , ther e exists a p ath fr om x to y , in S + t n , of length ≤ C e µd ( x,y ) . Pr o of. Let D b e the constan t of Prop osition 2.10, and let L > 0 for which ( ‡ 2 D ) holds. If d ( x, y ) ≤ 2 D , then prop ert y ( ‡ 2 D ) implies that there exists a path γ of length ≤ L from x to y in the complement of B n − D . By pro jecting bac k to S n an y sub-path of γ that lea ves S + t n , by item ( iii ) of Prop osition 2.11, there is a path from x to y in S + t n of length ≤ L (1 + 2 δ ) . Supp ose that d ( x, y ) > 2 D , and let γ b e a geo desic segment [ x, y ] . Set d := d ( x, y ) . Then γ lies in the complemen t of B ( e, n − d 2 − 1) . Let c, c ′ b e geo desic ra ys emanating from e , and p oin ts x ′ ∈ c and y ′ ∈ c ′ suc h that d ( x, x ′ ) ≤ D and d ( y , y ′ ) ≤ D, as in Proposition 2.10. Up to replacing D b y 3 D in the previous inequalities, we assume that x ′ , y ′ ∈ S n . Therefore, there exists a path γ ′ from x ′ to y ′ , of length ≤  := d + 2 L , that lies in the complemen t of B ( e, n − d 2 − 1) . In particular, γ ′ is a path b et ween c and c ′ . Therefore, there exists a path of length ≤ λ b etw een c to c ′ in the complement of B ( e, n − d 2 ) , where λ is the constan t from Lemma 2.15. W e can supp ose that λ > 1 . By 11 rep eating the pro cess ⌈ d 2 ⌉ times, we get a path α b et ween c and c ′ of length ≤ λ ⌈ d 2 ⌉  in the complemen t of B ( e, n − 1) . By item ( iii ) of Prop osition 2.11, projecting α to S n yields a path β from x ′ to y ′ in S + t n of length ≤ (1 + 2 δ ) λ ⌈ d 2 ⌉  . Since d ( x, x ′ ) ≤ 3 D , d ( y , y ′ ) ≤ 3 D , and 3 D ≤ t , there exists a path from x to y in S + t n of length ≤ (1 + 2 δ ) λ ⌈ d 2 ⌉ ( d + 2 L ) + 6 D . Since λ > 1 , there exists a constan t C ≥ 1 , dep ending only on δ , λ, L and D (and not on n or t ), such that for ev ery d ≥ 0 , (1 + 2 δ ) λ ⌈ d/ 2 ⌉ ( d + 2 L ) + 4 D ≤ C e µd , where µ := log( λ ) . This completes the pro of. Pr o of of Pr op osition 2.13. G is one-ended so δ > 0 . Let t ≥ t 0 b e fixed. Let n > t and x, y ∈ S + t n suc h that d ( x, y ) ≥ 9 δ + 6 t . Let x ′ , y ′ ∈ S n b e suc h that d ( x, x ′ ) ≤ t and d ( y , y ′ ) ≤ t. Then d ( x ′ , y ′ ) ≥ 9 δ + 4 t . By Lemma 2.14, d S + t n ( x ′ , y ′ ) ≥ 2 d ( x ′ ,y ′ ) 4 δ − 1 . Therefore, d S + t n ( x, y ) ≥ 2 d ( x,y ) − 2 t 4 δ − 1 − 2 t. Set µ 1 := log 2 4 δ and A 1 := 2 − t 2 δ − 1 . F or the upp er bound, we ha ve that d S + t n ( x ′ , y ′ ) ≤ C e µd ( x ′ ,y ′ ) , where µ, C > 0 are the constants from Lemma 2.16. Therefore, d S + t n ( x, y ) ≤ d S + t n ( x ′ , y ′ ) + 2 t ≤ C e µ ( d ( x,y )+2) + 2 t. Set A 2 := C e 2 µ + 2 t and µ 2 := µ . 2.2.3 Shado ws In the rest of this subsection, ∂ G is equipp ed with its visual metric ρ as in Section 2.2.1, and constan ts C 1 , C 2 as in Prop osition 2.8. Definition 2.17. If G is a non-elementary h yp erb olic group, let t 0 ≥ 0 b e as in Prop o- sition 2.11. F or x ∈ G , we define its shadow sh ( x ) ⊂ ∂ G to b e the set of b oundary p oin ts ξ for whic h there exists a geo desic ra y [ e, ξ ) passing through B ( x, t 0 ) . If A ⊂ G is a subset, we define its shado w by sh ( A ) := [ a ∈ A sh ( a ) . By the c hoice of t 0 , we hav e t 0 ≥ D , where D is the constan t from Prop osition 2.10. So for ev ery x ∈ G there exists x ′ ∈ B ( x, t 0 ) such that the geo desic segment [ e, x ′ ] extends to a geo desic ra y [ e, ξ ) . In particular, sh ( x )  = ∅ . Lemma 2.18. L et G b e a non-elementary hyp erb olic gr oup, t 0 as in Pr op osition 2.11, and t ≥ t 0 . Then for any n ∈ N , if x, y ∈ S + t n and d ( x, y ) ≥ 5 t 0 + 2 t , then sh ( x ) ∩ sh ( y ) = ∅ . 12 Pr o of. Let ξ ∈ sh ( x ) , and let x ′ ∈ B ( x, t 0 ) b e such that [ e, ξ ) extends [ e, x ′ ] . Let x ′′ b e the intersection of [ e, ξ ) with S n , so d ( x, x ′′ ) ≤ 2 t 0 + t . Let η ∈ sh ( y ) , and let y ′′ ∈ S n b e constructed similarly . So d ( x ′′ , y ′′ ) ≥ 5 t 0 + 2 t − 2(2 t 0 + t ) = t 0 . Then ρ ( ξ , η ) ≥ 1 C 1 a − n > 0 , where C 1 is the constant from Prop osition 2.8. A family of p oin ts { x i } i ∈ I in a metric space ( X, d ) is called r -sep ar ate d , for some r ≥ 0 , if d ( x i , x j ) ≥ r for ev ery distinct i, j ∈ I . Lemma 2.19. L et G b e a non-elementary hyp erb olic gr oup, and let t 0 b e as in Pr op o- sition 2.11. L et t ≥ t 0 , n ∈ N , ω > 0 , and Z ⊂ S + t n a subset. If ther e exists { ζ i } i ∈ I in sh ( Z ) ⊂ ∂ G a ω -sep ar ate d family, then ther e exist { z i } i ∈ I in Z a d ω -sep ar ate d family, with d ω := 2 n + 2 log a  ω C 2  − 4 t 0 − 2 t, wher e C 2 is the c onstant fr om Pr op osition 2.8 Pr o of. F or ev ery i ∈ I , ζ i ∈ sh ( Z ) , so there exists z i ∈ Z and z ′ i ∈ B ( z i , t 0 ) such that [ e, z ′ i ] extends to [ e, ζ i ) . Up taking the ball B ( z i , 2 t 0 + t ) instead of B ( z i , t 0 ) , w e can assume that z ′ i ∈ S n . F or every i  = j in I , ρ ( ζ i , ζ j ) ≥ ω , so b y (2), d ( z ′ i , z ′ j ) ≥ 2 n + 2 log a  ω C 2  . Since d ( z i , z ′ i ) ≤ 2 t 0 + t , w e get d ( z i , z j ) ≥ 2 n + 2 log a  ω C 2  − 4 t 0 − 2 t. Set d ω := 2 n + 2 log a  ω C 2  − 4 t 0 − 2 t . Prop osition 2.20. L et G b e a one-ende d hyp erb olic gr oup. L et t 0 b e as in Pr op osi- tion 2.11, and let t ≥ t 0 . Then ther e exists a c onstant θ > 0 such that the fol lowing holds. F or every n ∈ N , every subset Z ⊆ S + t n , and every p air of p oints x, y ∈ S + t n c ontaine d in differ ent c onne cte d c omp onents of S + t n \ Z and satisfying d ( { x, y } , Z ) ≥ θ, the shadows sh ( x ) and sh ( y ) ar e disjoint and lie in differ ent c onne cte d c omp onents of ∂ G \ sh ( Z ) . In p articular, sh ( Z ) sep ar ates ∂ G . Pr o of. Let A 1 , A 2 , µ 1 , µ 2 > 0 b e the constants as in Prop osition 2.13. Let θ ≥ 10 t b e large enough such that 2( A 1 e µ 1 θ − 2 t ) > A 2 e 10 tµ 2 . (6) W e sho w that the statemen t holds for this θ . If d ( x, Z ) ≥ θ , then d ( x, Z ) ≥ 10 t , and sh ( x ) and sh ( Z ) are disjoin t b y Lemma 2.18. The same holds for sh ( y ) and sh ( Z ) . Moreo ver, d ( x, Z ) ≥ θ implies that d S + t n ( x, Z ) ≥ A 1 e µ 1 d ( x,Z ) − 2 t ≥ A 1 e µ 1 θ − 2 t. 13 The same holds for y . Since x and y are in different connected comp onen ts of S + t n \ Z , w e obtain d S + t n ( x, y ) ≥ d S + t n ( x, Z ) + d S + t n ( y , Z ) ≥ 2( A 1 e µ 1 θ − 2 t ) . Therefore, b y Prop osition 2.13, d ( x, y ) ≥ log  d S + t n ( x,y ) A 2  µ 2 ≥ 7 t, where the righ t inequalit y holds b y (6). Therefore, sh ( x ) and sh ( y ) are disjoin t b y Lemma 2.18. No w suppose that there exists a path ˜ γ in ∂ G from sh ( x ) to sh ( y ) whic h a voids sh ( Z ) . Let x ′ (resp. y ′ ) b e the intersection of S n with a geodesic ray from e to sh ( x ) (resp. sh ( y ) ), and let γ be the pull-back of ˜ γ to S n giv en b y Prop osition 2.11. Then γ is a d 0 -path in S n from x ′ to y ′ whic h av oids Z + t 0 , where d 0 is the constant from Prop osition 2.8. Ho wev er, since d ( x, x ′ ) ≤ 2 t 0 + t < θ , the points x and x ′ lie in the same connected component of S + t n \ Z , and similarly y and y ′ lie in a same connected comp onen t. Hence x ′ and y ′ lie in t wo different connected components of S + t n \ Z . In particular, every path in S + t n from x ′ to y ′ in tersects Z and every d 0 -path from x ′ to y ′ m ust in tersect Z + d 0 , and in particular m ust in tersect Z + t 0 since t 0 ≥ d 0 . This con tradicts the fact that γ a voids Z + t 0 . 3 Pro of of the main theorem W e are now ready to pro ve Theorem 1.8. Pr o of of The or em 1.8. Let t 0 ≥ 1 be as in Proposition 2.11, and let t ≥ t 0 and ε ∈ (0 , 1) . Let n ∈ N , and let Z n ⊂ S + t n b e a ε -cut. Let Θ and α b e the constan ts from Prop osition 2.9. If the case (1) from Prop osition 2.9 do es not hold, then | Z n | ≥ Θ e α 2 n . (7) Otherwise, let x n , y n b e in differen t connected comp onen ts of S + t n \ Z n suc h that d ( { x n , y n } , Z n ) > n 2 . So, for n large enough, b y Prop osition 2.20, sh ( x n ) and sh ( y n ) are contained in tw o differen t connected comp onents of ∂ G \ sh ( Z n ) . Let ξ n ∈ sh ( x n ) , η n ∈ sh ( y n ) , and ζ ∈ sh ( Z n ) . So there exits z ∈ Z n , and z ′ ∈ B ( z , t 0 ) suc h that [ e, z ′ ] extends to [ e, ζ ) . Up taking the ball B ( z , 3 t ) instead of B ( z , t 0 ) , w e can assume that z ′ ∈ S n . Similarly , let x ′ n b e in B ( x n , 3 t ) ∩ S n suc h that [ e, x ′ n ] extends to [ e, ξ n ) . Since d ( x n , Z n ) > n 2 , n 2 ≤ d ( x n , z ) ≤ d ( x n , x ′ n ) + d ( x ′ n , z ′ ) + d ( z ′ , z ) ≤ d ( x ′ n , z ′ ) + 6 t. So d ( x ′ n , z ′ ) ≥ n 2 − 6 t , whic h is greater than d 0 (of Prop osition 2.8) for n large enough. Therefore, b y Prop osition 2.8, ρ ( ξ n , ζ ) ≥ C 1 a − n a n 2 − 6 t = C 1 a 6 t a − n 2 . 14 The same holds for ρ ( η n , ζ ) . Hence ρ ( { ξ n , η n } , sh ( Z n )) ≥ C 1 a 6 t a − n 2 . Set r 0 := C 1 a 6 t a − n 2 . By [LMM24, Claims 4.2 and 4.3], see Theorem 2.12, there exists a minimal closed subset, with resp ect to inclusion, Y ⊆ sh ( Z n ) that separates ξ n , η n . In particular ρ ( { ξ n , η n } , Y ) ≥ r 0 . Moreo ver, there exists a constan t λ ∈ (0 , 1) , that only dep ends on ( ∂ G, ρ ) , suc h that for all ζ ∈ Y and r ∈ (0 , r 0 ) , w e hav e Y ∩  ¯ B ρ ( ζ , r ) \ B ρ ( ζ , λr )   = ∅ , (8) where ¯ B ρ denotes the closed ball in ( ∂ G, ρ ) , and B ρ denotes the op en ball. Since λ < 1 , up to replacing it by λ N for some N ∈ N , w e may assume that λ < 1 3 . Cho ose some ζ 1 ∈ Y , and fix r = r 0 2 . Claim 3.1. F or every p ∈ N ∗ , ther e exist 2 p p oints in Y that ar e λ 2 p r -sep ar ate d. Pr o of of the claim. W e prov e it b y induction on p . F or p = 1 , by (8), there exists ζ 2 ∈ Y ∩  ¯ B ρ ( ζ 1 , λr ) \ B ρ ( ζ 1 , λ 2 r )  . So { ζ 1 , ζ 2 } are λ 2 r -separated. Supp ose that w e found, 2 p p oin ts { ζ i } 1 ≤ i ≤ 2 p in Y that are λ 2 p r -separated. Note that, since λ < 1 3 , for any i  = j in { 1 , . . . , 2 p } , B ρ ( ζ i , λ 2 p +1 r ) ∩ B ρ ( ζ j , λ 2 p +1 r ) = ∅ b ecause 2 λ 2 p +1 r < λ 2 p r ≤ ρ ( ζ i , ζ j ) . By applying (8), for ev ery i ∈ { 1 , . . . , 2 p } , there exists ζ ′ i ∈ Y ∩  ¯ B ρ ( ζ i , λ 2 p +1 r ) \ B ρ ( ζ i , λ 2 p +2 r )  . F or any i  = j , B ρ ( ζ i , λ 2 p +1 r ) ∩ B ρ ( ζ j , λ 2 p +1 r ) = ∅ , so ρ ( ζ ′ i , ζ j ) ≥ λ 2 p +1 r > λ 2 p +2 r . Moreo ver, λ 2 p r ≤ ρ ( ζ i , ζ j ) ≤ ρ ( ζ i , ζ ′ i ) + ρ ( ζ ′ i , ζ ′ j ) + ρ ( ζ ′ j , ζ j ) ≤ λ 2 p +1 r + ρ ( ζ ′ i , ζ ′ j ) + λ 2 p +1 r . Therefore, ρ ( ζ ′ i , ζ ′ j ) ≥ λ 2 p r − 2 λ 2 p +1 r = λ 2 p r (1 − 2 λ ) > λ 2 p +2 r , where the right inequalit y holds because λ < 1 3 . Therefore, { ζ i , ζ ′ i } 1 ≤ i ≤ 2 p in Y are λ 2( p +1) r -separated. By Lemma 2.19, ev ery λ 2 p r -separated family in Y ⊂ sh ( Z n ) yields a d p -separated family in Z n , where d p := 2 n + 2 log a λ 2 p r C 2 ! − 4 t 0 − 2 t. 15 Therefore, to get distinct p oin ts in G , one should hav e d p ≥ 1 . Since r = r 0 2 = C 1 2 a 6 t a − n 2 , w e hav e d p ≥ 1 ⇐ ⇒ 2 n + 2 log a λ 2 p C 1 2 a 6 t a − n 2 C 2 ! − 4 t 0 − 2 t ≥ 1 ⇐ ⇒ p ≤ τ n + Ω , where τ := 1 4 log a  1 λ  and Ω := 2 log a  C 1 2 C 2  − 14 t − 4 t 0 − 1 4 log a  1 λ  . Note that τ and Ω only depend on G and t , and that τ > 0 b ecause λ < 1 . Therefore p can b e taken p = ⌊ τ n + Ω ⌋ ≥ τ n + Ω − 1 . So Z n con tains a family of at least 2 Ω − 1 2 τ n p oin ts that are 1 -separated. Therefore | Z n | ≥ 2 Ω − 1 2 τ n . Com bining this with (7), we conclude that | Z n | ≥ min { 2 Ω − 1 2 τ n , Θ e α 2 n } . Pr o of of The or em 1.1. If G splits o v er a t wo-ended subgroup H , then G is coarsely sep- arated b y H . Moreov er, since H is virtually cyclic, and cyclic subgroups are undistorted in h yp erbolic groups, then H is undistorted. Therefore G is coarsely separated b y a subset of linear growth. Supp ose that G do es not split o ver a t wo-ended subgroup, and let t ≥ 0 b e as in Theorem 1.8. Since the family { S G ( g , n ) + t } g ∈ G, n ≥ t is persistent (see Example 2.6), it follo ws from Theorem 2.7 that G cannot b e coarsely separated b y an y family of subsets with sub-exp onen tial growth. 4 Applications to graph pro ducts This section is dedicated to the pro of of Theorem 1.10. W e start b y recalling basic defi- nitions and prop erties related to graph pro ducts in S ection 4.1. Section 4.2 is dedicated to the h yp erb olicit y of suc h groups. In particular, w e determine precisely when a graph pro duct is a virtual surface group. In Section 4.3, w e study splittings of graph pro ducts, and in particular we characterise splittings o ver virtually cyclic groups. Finally , we pro ve Theorem 1.10 in Section 4.4, where we also men tion a couple of concrete examples. 4.1 Graph pro ducts Giv en a graph Γ and a collection of groups G = { G u | u ∈ V (Γ) } indexed b y the vertices of Γ , the gr aph pr o duct Γ G is the group defined by the relativ e presentation ⟨ G u ( u ∈ V (Γ)) | [ G u , G v ] = 1 ( { u, v } ∈ E (Γ)) ⟩ , where E (Γ) denotes the edge-set of Γ and where [ G u , G v ] = 1 is a shorthand for “ [ a, b ] = 1 for all a ∈ G u and b ∈ G v . W e refer to the groups in G as the vertex-gr oups . Usually , one says that graph products interpolate b etw een free pro ducts (when Γ has no edge, i.e. “nothing comm ute”) and direct sums (when Γ is a complete graph, i.e. “ev erything comm ute”). Classical examples also include right-angled Artin groups (= graph products of infinite cyclic groups) and righ t-angled Co xeter groups (= graph pro ducts of cyclic groups of order 2 ). Con ven tion. In the rest of the article, we will alw a ys assume that the vertex-groups of our graph pro ducts are non-trivial. 16 Giv en a graph Γ and a collection of groups G indexed by V (Γ) , an elemen t g ∈ Γ G can b e written as a pro duct s 1 · · · s n where eac h s i b elongs to a v ertex-group. W e refer to suc h a pro duct as a wor d and to each s i as a syl lable . Notice that, giv en such a w ord r 1 · · · r m , applying the follo wing op erations do es not mo dify the element of Γ G it represen ts: (Cancellation) if there exists some 1 ≤ i ≤ m suc h that r i = 1 , remov e the syllable r i ; (Merging) if there exists some 1 ≤ i ≤ m − 1 such that r i and r i +1 b elong to the same v ertex-group, replace the sub word r i r i +1 with the single syllable ( r i r i +1 ) ; (Sh uffling) if there exists some 1 ≤ i ≤ m − 1 such that r i and r i +1 b elong to adjacen t v ertex-groups, replace the sub word r i r i +1 with the subw ord r i +1 r i . A word is gr aphic al ly r e duc e d if it cannot b e shortened b y applying a sequence of such mo ves. Prop osition 4.1 ([Gre90]) . L et Γ b e a gr aph and G a c ol le ction of gr oups indexe d by V (Γ) . Every element of Γ G c an b e r epr esente d by a gr aphic al ly r e duc e d wor d. Mor e over, any two such wor ds only differ fr om e ach other by a se quenc e of shufflings. In the follo wing, we will need some v o cabulary related to graphs, whic h we introduce no w. • A join A ∗ B of tw o graphs A and B is the graph obtained from the disjoin t union A ⊔ B b y connecting with an edge every vertex of A to every vertex of B . • Giv en a graph Γ and a vertex u ∈ V (Γ) , the link of u in Γ , denoted link( u ) , is the subgraph of Γ induced by the neigh b ours of u . • Giv en a graph Γ and a vertex u ∈ V (Γ) , the star of u in Γ , denoted star( u ) , is the subgraph of Γ induced by link( u ) ∪ { u } . Finally , giv en a graph Γ , a collection of groups G indexed by V (Γ) , and a subgraph Λ ≤ Γ , w e denote b y ⟨ Λ ⟩ the subgroup of Γ G generated by the v ertex-groups indexed b y the v ertices of Λ . Notice that, if Λ decomp oses as a join Φ ∗ Ψ , then ⟨ Λ ⟩ = ⟨ Φ ⟩ ⊕ ⟨ Ψ ⟩ . 4.2 Hyp erb olicit y In this section, w e fo cus on the hyperb olicit y of graph pro ducts. First of all, we record the follo wing characterisation: Theorem 4.2 ([Mei96]) . L et Γ b e a finite gr aph and G a c ol le ction of finitely gener ate d gr oups indexe d by V (Γ) . The gr aph pr o duct Γ G is hyp erb olic if and only if the fol lowing c onditions hold: • every gr oup of G is hyp erb olic; • no two infinite vertex-gr oups ar e adjac ent in Γ ; • two vertex-gr oups adjac ent to a c ommon infinite vertex-gr oup must b e adjac ent; • the gr aph Γ is □ -fr e e. No w, w e w ould lik e to identify when a given graph pro duct belong to some specific families of hyperb olic groups. 17 When virtually cyclic. W e first consider the family of virtually cyclic groups. Prop osition 4.3. L et Γ b e a finite gr aph and G a c ol le ction of gr oups indexe d by V (Γ) . The gr aph pr o duct Γ G is virtual ly cyclic if and only if • either Γ is a c omplete gr aph al l of whose vertic es ar e lab el le d by finite gr oups; • or Γ is a c omplete gr aph with one vertex lab el le d by a virtual ly infinite cyclic gr oup and al l the other vertic es lab el le d by finite gr oups; • or Γ is a join b etwe en a c omplete gr aph al l of whose vertic es ar e lab el le d by finite gr oups and two non-adjac ent vertic es lab el le d by Z 2 . Our pro of of the prop osition will be based on the following observ ation: Lemma 4.4. L et Γ b e a gr aph, G a c ol le ction a gr oup indexe d by V (Γ) , and Λ ≤ Γ a sub gr aph. The sub gr oup ⟨ Λ ⟩ has finite index in Γ G if and only if Γ = star(Λ) and link (Λ) is a finite c omplete gr aph al l of whose vertic es ar e lab el le d by finite gr oups. Pr o of. If Γ = star(Λ) and link(Λ) is a finite complete graph all of whose v ertices are lab elled by finite groups, then Γ G = ⟨ Λ ⟩ ⊕ ⟨ link (Λ) ⟩ where ⟨ link (Λ) ⟩ is finite (as a pro duct of finitely many finite groups). Consequen tly , ⟨ Λ ⟩ has finite index in Γ G . Con versely , assume that ⟨ Λ ⟩ has finite index in Γ G . If Γ  = star(Λ) , then w e can find t wo vertices u / ∈ V (link (Λ)) and v ∈ V (Λ) that are not adjacen t. Fix t wo non-trivial elemen ts a ∈ ⟨ u ⟩ and b ∈ ⟨ v ⟩ . By noticing that ( ab ) n do es not belong to ⟨ Λ ⟩ for ev ery n ≥ 1 , we deduce that ⟨ Λ ⟩ cannot hav e finite index in Γ G . Consequen tly , we m ust ha ve Γ = star(Λ) . This implies that Γ G = ⟨ Λ ⟩ ⊕ ⟨ link(Λ) ⟩ . Clearly , ⟨ Λ ⟩ has finite index in Γ G if and only if ⟨ link (Λ) ⟩ is finite, whic h holds if and only if link(Λ) is a finite complete graph all of whose vertices are labelled by finite groups. Pr o of of Pr op osition 4.3. Decomp ose Γ as a join Γ 0 ∗ Γ 1 ∗ · · · ∗ Γ n where Γ 0 is a complete graph (p ossibly empt y) and where Γ 1 , . . . , Γ n are not joins and eac h contains at least t wo v ertices. Notice that Γ G = ⟨ Γ 0 ⟩ ⊕ ⟨ Γ 1 ⟩ ⊕ · · · ⊕ ⟨ Γ n ⟩ ; and that, for every 1 ≤ i ≤ n , ⟨ Γ i ⟩ is infinite since Γ i is not complete. Th us, if n ≥ 2 , then Γ G cannot b e virtually cyclic since it would contain a pro duct of t wo infinite groups. Similarly , if n = 1 and Γ 0 has a vertex lab elled b y an infinite group, then Γ G cannot b e virtually cyclic. So only tw o cases remain to b e considered. The first case is n = 0 . In this case, Γ is a complete graph, whic h amoun ts to saying that Γ G is the direct sum of its v ertex-groups. Thus, Γ G is virtually cyclic if and only if either v ertex-groups are all finite (i.e. Γ is a complete graph all of whose vertices are lab elled by finite groups) or there is exactly one v ertex-group that is virtually infinite cyclic while all the others are finite (i.e. Γ is a complete graph with one v ertex lab elled b y a virtually infinite cyclic group and all the other vertices labelled b y finite groups). The second case is n = 1 and Γ 0 is a complete graph all of whose vertices are lab elled b y finite groups. Then, Γ G = ⟨ Γ 0 ⟩ ⊕ ⟨ Γ 1 ⟩ with ⟨ Γ 0 ⟩ finite, so Γ G is virtually cyclic if and only if ⟨ Γ 1 ⟩ is virtually cyclic. Because Γ 1 is not complete and contains at least t w o v ertices, it must contain t wo non-adjacen t v ertices u, v ∈ V (Γ 1 ) . Since ⟨ u, v ⟩ ≃ ⟨ u ⟩ ∗ ⟨ v ⟩ is necessarily virtually cyclic, the groups labelling u and v m ust b e b oth Z 2 . Then, ⟨ u, v ⟩ is an infinite dihedral group; and, since it must hav e finite index in ⟨ Γ 1 ⟩ , w e deduce from Lemma 4.4 and from the fact that Γ 1 is not a join, that Γ 1 is reduced to the pair of non-adjacen t vertices { u, v } . Th us, w e hav e pro ved that Γ is a join betw een a finite complete graph all of whose v ertices are lab elled b y finite groups (namely , Γ 0 ) and tw o non-adjacen t vertices labelled b y Z 2 (namely , u and v ). 18 When a surface group. Then, w e would like to determine when a graph pro duct is virtually a surface group. Here, we refer to a surface group as the fundamen tal group of a closed surface of genus ≥ 2 . Prop osition 4.5. L et Γ b e a finite gr aph and G a c ol le ction of gr oups indexe d by V (Γ) . The gr aph pr o duct Γ G is virtual ly a surfac e gr oup if and only if • either Γ is a c omplete gr aph with one vertex lab el le d by a virtual surfac e gr oup and al l the other vertic es by finite gr oups; • or Γ de c omp oses as a join b etwe en a c omplete gr aph al l of whose vertic es ar e lab el le d by finite gr oups and a cycle of length ≥ 5 al l of whose vertic es ar e lab el le d by Z 2 . Before turning to the pro of of our characterisation, we observ e that: Prop osition 4.6. L et Γ b e a finite gr aph and G , H two c ol le ctions of finitely gener ate d gr oups indexe d by V (Γ) . If ther e exists a biLipschitz emb e dding ι u : G u  → H u for every u ∈ V (Γ) , then ther e exists a biLipschitz emb e dding Γ G  → Γ H . The pro of of our prop osition will be based on the following technical lemma: Lemma 4.7. L et Γ b e a gr aph and G a c ol le ction of gr oups indexe d by V (Γ) . A ny two elements x, y ∈ Γ G c an b e r epr esente d r esp e ctively as gr aphic al ly r e duc e d wor ds pa 1 · · · a n r and pb 1 · · · b n s such that s − 1 ( a 1 b − 1 1 ) · · · ( a n b − 1 n ) r is a gr aphic al ly r e duc e d wor d r epr esenting y − 1 x . Pr o of. W e prov e our lemma by using the geometric p erspective given b y [Gen17]. W e kno w from [Gen17, Prop osition 8.2] that the Cayley graph QM(Γ , G ) := Cayl   Γ G , [ G ∈G G   is quasi-median, so it follo ws from [Gen17, Prop osition 2.84] that there exist an equilat- eral triangle p, pq x , pq y ∈ Γ G such that equalities      d (1 , x ) = d (1 , p ) + d ( p, pq x ) + d ( pq x , x ) d (1 , y ) = d (1 , p ) + d ( p, pq y ) + d ( pq y , y ) d ( x, y ) = d ( x, pq x ) + d ( pq x , pq y ) + d ( pq y , y ) hold in QM(Γ , G ) . Since geodesics in QM(Γ , G ) are given b y graphically reduced w ords [Gen17, Lemma 8.3], the first (resp. second) equalit y sho ws that x (resp. y ) can be written as a graphically reduced w ord pq x r (resp. pq y s ). Then, the third equalit y sho ws that y − 1 x can b e written as a graphically reduced w ord s − 1 q r where q is a graphically reduced word representing q − 1 y q x . But we know from [Gen17, Proposition 8.2 and Corollary 8.7] that q x and q y b elong to ⟨ Λ ⟩ for some complete subgraph Λ ≤ Γ , so w e can write q x and q y resp ectiv ely as graphically reduced w ords a 1 · · · a n and b 1 · · · b m of pairwise comm uting syllables. Notice that, since our triangle p, pq x , pq y is equilateral, necessarily n = m and ( b − 1 1 a 1 ) · · · ( b − 1 n a n ) is graphically reduced. The desired conclusion follo ws. Pr o of of Pr op osition 4.6. F or con venience, let ι : S G  → S H denote the map that restricts to ι u on each G u . Giv en an elemen t g ∈ Γ F , w e write g as a graphically reduced w ord s 1 · · · s n and w e define Φ( g ) := ι ( s 1 ) · · · ι ( s n ) . Notice that, as a consequence of Prop osition 4.1, the element of Γ H that Φ( g ) defines do es not dep end on the choice of 19 the graphically reduced w ord represen ting g . W e claim that the map Φ : Γ G → Γ H is a biLipsc hitz embedding. So let x, y ∈ Γ G b e t wo elemen ts. W rite x and y resp ectively as graphically reduced w ords pa 1 · · · a n r and pb 1 · · · b n s as given by Lemma 4.7. Let r 1 . . . r k (resp. s 1 · · · s ℓ ) be a graphically reduced word representing r (resp. s ). On the one hand, d ( x, y ) = ℓ X i =1 ∥ s i ∥ + n X i =1 ∥ b − 1 i a i ∥ + k X i =1 ∥ r i ∥ , where ∥ · ∥ denotes the w ord-length of the syllable under consideration inside the v ertex- group that contains it. On the other hand, d (Φ( x ) , Φ( y )) = ℓ X i =1 ∥ ι ( s i ) ∥ + n X i =1 ∥ ι ( b i ) − 1 ι ( a i ) ∥ + k X i =1 ∥ ι ( r i ) ∥ , hence d (Φ( x ) , Φ( y )) ≥ 1 L ℓ X i =1 ∥ s i ∥ + n X i =1 ∥ b − 1 i a i ∥ + k X i =1 ∥ r i ∥ ! = 1 L d ( x, y ) and d (Φ( x ) , Φ( y )) ≤ L ℓ X i =1 ∥ s i ∥ + n X i =1 ∥ b − 1 i a i ∥ + k X i =1 ∥ r i ∥ ! = Ld ( x, y ) , where L > 0 is a constant suc h that each ι u is L -biLipsc hitz. Pr o of of Pr op osition 4.5. Decomp ose Γ as a join Γ 0 ∗ Γ 1 ∗ · · · ∗ Γ n where Γ 0 is a complete graph and where Γ 1 , . . . , Γ n are not joins and eac h con tains at least tw o v ertices. Notice that, for ev ery 1 ≤ i ≤ n , ⟨ Γ i ⟩ is infinite since Γ i is not complete and contains at least t wo vertices. Therefore, if n ≥ 2 , then Γ G con tains a pro duct of t wo infinite groups and consequen tly cannot b e h yp erb olic (and a fortiori not a virtual surface group). If n = 0 , then Γ = Γ 0 is complete and Γ G coincides with the pro duct of its vertex-groups. In order to b e a virtual surface group, again b ecause a virtual surface group (and more generally an y h yp erbolic group) do es not con tain a pro duct of t wo infinite groups, exactly one v ertex of Γ m ust b e lab elled by a virtual surface group and all the other v ertices m ust be lab elled by finite groups. It remains to consider the case n = 1 . Since Γ G = ⟨ Γ 0 ⟩ ⊕ ⟨ Γ 1 ⟩ with ⟨ Γ 1 ⟩ infinite, necessarily ⟨ Γ 0 ⟩ must b e finite, i.e. all the v ertices of Γ 0 m ust be lab elled b y finite groups. Notice that, then, Γ G is a virtual surface group if and only if ⟨ Γ 1 ⟩ is virtually a surface group. Th us, in order to conclude the pro of of our prop osition, it remains to v erify that, if Γ is not a join and con tains at least t wo vertices, then Γ G is a virtual surface group if and only if Γ is a cycle of length ≥ 5 all of whose v ertices are lab elled by Z 2 . If Γ is c hordal, i.e. it do es not con tain an induced cycle of length ≥ 4 , then w e kno w from [Dir61] that it con tains a separating complete subgraph, say Λ ≤ Γ . W e c ho ose Λ with the least num b er of vertices. If all the vertices of Λ are lab elled by finite groups, then Γ G splits ov er a finite subgroup (namely , ⟨ Λ ⟩ ), which implies that Γ G is multi- ended, and consequen tly not a virtual surface group. So there exists u ∈ V (Λ) suc h that ⟨ u ⟩ is infinite. If link( u ) is not complete, i.e. there exist tw o non-adjacent vertices v , w adjacen t to u , then Γ G cannot b e hyperb olic (and a fortiori not a virtual surface group) since it contains a product of t w o infinite groups, namely ⟨ u, v , w ⟩ ≃ ⟨ u ⟩ ⊕ ( ⟨ v ⟩ ∗ ⟨ w ⟩ ) . So link( u ) must b e complete. But then Λ \{ u } must also separate Γ , con tradicting the minimalit y of Λ . Th us, we hav e prov ed that Γ cannot b e c hordal. In other w ords, Γ must contain a cycle of length ≥ 4 . If ( r , s, t, u ) is a cycle of length 4 in Γ , then Γ G cannot b e hyperb olic (and a fortiori not a virtual surface group) since 20 it con tains a pro duct of tw o infinite groups, namely ⟨ r , s, t, u ⟩ = ( ⟨ r ⟩ ∗ ⟨ t ⟩ ) ⊕ ( ⟨ s ⟩ ∗ ⟨ u ⟩ ) . Th us, Γ must con tain a cycle Ξ of length ≥ 5 . First, assume that all the vertices of Ξ are labelled b y Z 2 . Then, ⟨ Ξ ⟩ is a virtual surface group. Since infinite-index subgroups in surface groups are free, it follo ws that Γ G is a virtual surface group if and only if ⟨ Ξ ⟩ has finite index in Γ G . A ccording to Lemma 4.4, and b ecause Γ is not a join, this amoun ts to saying that Γ = Ξ . In other words, Γ is a cycle of length ≥ 5 all of whose v ertices are lab elled b y Z 2 . Next, assume that at least one v ertex of Ξ is lab elled by a group of size ≥ 3 . As a consequence of Prop osition 4.6, there exists a quasi-isometric embedding Ξ H → Γ G where H is a collection with one Z 3 while all its other groups are Z 2 . In order to conclude the pro of of our prop osition, w e need to verify that Γ G is not a virtual surface group, whic h is a consequence of the following observ ation: Claim 4.8. The gr oup Ξ H is a one-ende d hyp erb olic gr oup that is not virtual ly a surfac e gr oup. W e kno w from Theorem 4.2 and [V ar20] (see also Corollary 4.10 b elo w) that Ξ H is a one-ended hyperb olic group. It follows from [Kim12, Lemma 21] that Ξ H con tains a subgroup isomorphic to the graph pro duct Θ(2) of cyclic groups Z 2 (i.e. a right-angled Co xeter group) where Θ is the union of three paths of length (length(Ξ) − 2) glued along their endp oints. T w o suc h paths yield a cycle Ω of length 2(length(Ξ) − 2) ≥ 6 . Thus, Θ(2) con tains a virtual surface subgroup, namely ⟨ Ω ⟩ , whic h has infinite index according to Lemma 4.4. Since infinite-index subgroups in surface groups are free, it follows that Θ(2) , and a fortiori Ξ H , is not a virtual surface group. 4.3 Splittings The last step tow ards the proof of Theorem 1.10 is to understand when a graph product of finite groups splits o ver a virtually cyclic group. Our characterisation will follo w from the our next observ ation, which follows the lines of [GH17]. Given a group G and a collection of subgroups H , w e refer to a splitting of G r elative to H as a splitting of G for which every subgroup in H is elliptic in the corresp onding Bass-Serre tree (or equiv alently , is con tained in a conjugate of a factor of the decomp osition). Prop osition 4.9. L et Γ b e a finite gr aph, G a c ol le ction of gr oups indexe d by V (Γ) , and H ≤ Γ G a sub gr oup. If Γ G splits over H r elative to G , then ther e exists a sep ar ating sub gr aph Λ ≤ Γ such that ⟨ Λ ⟩ has a c onjugate c ontaine d in H . Pr o of. W e kno w that Γ G acts non-trivially on some tree T with edge-stabilisers conjugate to H and with elliptic v ertex-groups, namely the Bass-Serre tree given by our splitting. F or ev ery v ertex u ∈ V (Γ) , consider the non-empty subtree T u := Fix ( ⟨ u ⟩ ) . If the T u pairwise intersect, then they must globally intersect. In other words, we find a p oin t globally fixed by Γ G , but w e kno w that it is not the case. Consequently , there exist u, v ∈ V (Γ) such that T u ∩ T v = ∅ . Fix an edge e of the geo desic connecting T u and T v and let Λ denote the subgraph of Γ induced b y the vertices w ∈ V (Γ) for which ⟨ w ⟩ fixes e (or equiv alently , e ∈ E ( T w ) ). Since T a ∩ T b  = ∅ if a, b ∈ V (Γ) are adjacent, necessarily Λ is a non-empt y subgraph that separates u and v in Γ . Since ⟨ Λ ⟩ is contained in stab( e ) , whic h is conjugate to H , the desired conclusion follows. Corollary 4.10. L et Γ b e a finite gr aph and G a c ol le ction of finite gr oups indexe d by V (Γ) . The gr aph pr o duct Γ G • splits over a finite sub gr oup, i.e. is multi-ende d, if and only if Γ c ontains a sep a- r ating c omplete sub gr aph; 21 • splits over a virtual ly cyclic sub gr oup if and only if it c ontains a sep ar ating sub gr aph that is a join A ∗ B b etwe en a c omplete gr aph A and a gr aph B that is either empty or two non-adjac ent vertic es lab el le d by Z 2 . Pr o of. If Γ G splits ov er a finite subgroup, then we kno w from Proposition 4.9 that Γ con tains a separating subgraph Λ such that ⟨ Λ ⟩ is finite. Necessarily , Λ m ust be a complete subgraph all of whose vertices are lab elled by finite groups. Conv ersely , it is clear that, if w e kno w that Γ con tains such a subgraph, then Γ G splits o ver a finite subgroup. If Γ G splits o ver a virtually cyclic group, then w e know from Prop osition 4.9 that Γ con tains a separating subgraph Λ suc h that ⟨ Λ ⟩ is virtually cyclic. A ccording to Prop o- sition 4.3, Λ is either a complete graph all of whose vertices are labelled by finite groups or a join b et ween a complete graph all of whose v ertices are labelled by finite groups and t wo non-adjacent v ertices lab elled b y Z 2 . Con v ersely , it is clear that, if we kno w that Γ contains such a subgraph, then Γ G splits o ver a virtually cyclic subgroup. 4.4 Pro of of Theorem 1.10 and examples W e are finally ready to pro ve our main result ab out h yp erbolic graph pro ducts of finite groups. Pr o of of The or em 1.10. If Γ contains a separating subgraph Λ = Λ 1 ∗ Λ 2 where Λ 1 is a complete graph all of whose vertices are lab elled by finite groups and where Λ 2 is either empt y or t w o non-adjacen t v ertices both lab elled by Z 2 , then Γ G splits ov er ⟨ Λ ⟩ , whic h is virtually cyclic. This implies that Γ G is coarsely separable b y a family of sub exponential gro wth. Con versely , assume that Γ G is coarsely separable by a family of sub exp onen tial growth. Notice that w e know from Theorem 4.2 that Γ G is h yp erb olic. Moreo v er, Corollary 4.10 sho ws that Γ G is m ulti-ended if and only if Γ contains a separating subgraph all of whose v ertices are lab elled by finite groups; and Prop osition 4.5 shows that Γ G is virtually a surface group if and only if Γ decomposes as a join b et ween a complete graph all of whose v ertices are lab elled by finite groups and a cycle of length ≥ 5 all of whose vertices are lab elled b y Z 2 . In the latter case, notice that Γ contains a separating subgraph that is a join b et ween a complete graph all of whose v ertices are lab elled by finite groups and t wo non-adjacen t vertices lab elled by Z 2 . F rom now on, assume that Γ G is one-ended and is not virtually a surface group. Then, w e kno w from Theorem 1.1 that Γ G splits o v er a virtually cyclic group. Then, the desired conclusion if provided by Corollary 4.10. W e conclude this section with a few concrete examples. First, as an immediate conse- quence of Theorem 1.10: Corollary 4.11. L et Γ b e a finite □ -fr e e gr aph. The right-angle d Coxeter gr oup C (Γ) is c o arsely sep ar able by a family of sub exp onential gr owth if and only if Γ c ontains a sep ar ating sub gr aph A ∗ B wher e A is c omplete and wher e B is either c omplete or c onsists of a p air of non-adjac ent vertic es. Our next examples will pla y an imp ortan t role in our forthcoming w ork [BGT26], dedi- cated to coarse separation in righ t-angled Artin groups. They also include the so-called Bourdon groups, hence extending our previous result [BGT24, Theorem 1.3]. Example 4.12. Let Γ b e a b e a cycle of length ≥ 5 and G a collection of finite groups indexed by V (Γ) . The graph pro duct Γ G is coarsely separable b y a family of sub ex- p onen tial growth if and only if Γ has tw o non-adjacen t v ertices that are both labelled b y Z 2 . 22 Figure 1: The tw o graph pro ducts from Example 4.13. Finally , let us illustrate the fact that Theorem 1.10 can ha v e concrete applications to the problem of determining whether or not there exist coarse embeddings b etw een finitely generated groups. Example 4.13. Let G 1 and G 2 b e the tw o graph products illustrated by Figure 1. There is no coarse em b edding G 1 → G 2 . Indeed, as indicated on the figure, G 2 can b e decomp osed as an amalgamated pro duct o ver an infinite dihedral group of t wo smaller graph pro ducts, whic h turn out to be virtually free according to [LS07]. 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UCLouv ain Institut de recherche en Ma théma tiques et physique Chemin du Cyclotron 2 1348 Louv ain-la-Neuve (Belgium) E-mail addr ess : oussama.bensaid@uclouvain.be University of Montpellier Institut Ma théma tiques Alexander Grothendieck Place Eugène Ba t aillon 34090 Montpellier (France) E-mail addr ess : anthony.genevois@umontpellier.fr University of P aris-Cité Institut de Ma théma tiques de Jussieu-P aris Rive Gauche Place A urélie Nemours 75013 P aris (France) E-mail addr ess : romain.tessera@imj-prg.fr 24